CP and Lepton-Number Violation in GUT Neutrino Models with Abelian Flavour Symmetries

34 Pages·2007·0.42 MB·English

Published by Institute of Physics Publishing for SISSA
Received: January 24, 2007
Revised: June 9, 2007
Accepted: July 5, 2007
Published: July 18, 2007
CP and lepton-number violation in GUT neutrino
models with abelian flavour symmetries
J
John Ellis
H
Theory Division, CERN,
E
CH-1211 Geneva 23, Switzerland
E-mail: [email protected] P
0
Mario E. G´omez
7
Departamento de F´ısica Aplicada, University of Huelva,
Facultad de Ciencias Experimentales, Campus El Carmen, 21071 Huelva, Spain (
E-mail: [email protected]
2
Smaragda Lola 0
Department of Physics, University of Patras, 0
Panepistimiopolis, GR-26500 Patras, Greece
7
E-mail: [email protected]
)
Abstract: We study the possible magnitudes of CP and lepton-number-violating quan- 0
tities in specific GUT models of massive neutrinos with different Abelian flavour groups, 5
taking into account experimental constraints and requiring successful leptogenesis. We 2
discuss SU(5) and flipped SU(5) models that are consistent with the present data on neu-
trino mixingandupperlimits onthe violations of charged-lepton flavours and exploretheir
predictions for the CP-violating oscillation and Majorana phases. In particular, we discuss
string-derived flipped SU(5) models with selection rules that modify the GUT structure
and provide additional constraints on the operators, which are able to account for the
magnitudes of some of the coefficients that are often set as arbitrary parameters in generic
Abelian models.
Keywords: Supersymmetry Phenomenology, Rare Decays, Neutrino Physics.
°c SISSA2007 http://jhep.sissa.it/archive/papers/jhep072007052/jhep072007052.pdf
Contents
1. Introduction 1
2. Observables in the lepton sector 4
2.1 Neutrino observables 4
2.2 Leptogenesis 5
2.3 Charged-lepton-flavour violation 7
3. GUTs with abelian flavour symmetries 8
3.1 SU(5) 9
J
3.2 An example with large tanβ 10
3.3 Examples with small tanβ 15 H
3.4 Flipped SU(5) model 20 E
3.5 String-derived flipped SU(5) models 22
P
4. Conclusions 27 0
7
A. GUT Fits 2 and 4 of [32] 27
(
B. Flipped SU(5) particle assignments 28 2
0
0
7
1. Introduction )
0
In recent years, several experiments have provided convincing evidence for neutrino oscil-
5
lations [1–5], together with important information on the neutrino mass differences and
2
mixing angles [6]. It has now been established that the atmospheric neutrino mixing angle,
θ , is large, witha central value π/4, sothat 0.31 sin2θ 0.72. Thesolar MSWan-
23 23
∼ ≤ ≤
gleθ [7]isalsolarge,butnotmaximal,withsin2θ 0.3. UpperlimitsfromtheCHOOZ
12 12
∼
experiment, in particular, establish that the third mixing angle, θ , must be small, with a
13
presentupperlimitof sin2θ 0.1 [8]. Finally, experimentshave established thefollowing
13
≤
ranges for the mass differences: ∆m2 2.5 10−3 eV2 and ∆m2 8 10−5 eV2 [6].
atm ∼ sol ∼
These important results leave several questions open for model builders to consider:
Is the atmospheric neutrino mixing indeed maximal?
•
Howlargeisthesolarneutrinomixing,andcanitberelatedtomixingamongquarks?
•
What is the pattern of masses for the neutrinos?
•
What is the magnitude of θ ?
13
•
– 1 –
Is there significant CP violation in the neutrino and charged-lepton sectors?
•
Inthispaperwestudy,inparticular,theanswerstothelasttwoquestionsthatareprovided
by models that propose different answers to the first three questions.
The existence of neutrino masses and oscillations may have further consequences. For
example, thedecays ofheavy neutrinosmay lead toleptogenesis, whichwouldimposeaddi-
tionalconstraintsonCP-violatingphasesandonthemagnitudesofcertainneutrinoYukawa
couplings [9]. These should be large enough to generate a sufficient net lepton number,
but not above the value that would erase any generated asymmetry through equilibriation
effects. Also, mixing in the charged-lepton sector is enhanced by radiative corrections in
supersymmetric models that mix the sleptons, contributing via loop diagrams to rare de-
cays and flavour conversions [10, 11]. The rates for such processes mediated by sparticles
may be large and close to the current experimental bounds in many supersymmetric mod-
els. Moreover, the possibility of CP-violation is crucial in this analysis [12] and additional J
CP-violating quantities may beobservableinthecharged-lepton sector. Themodelparam- H
eters thatwouldberesponsiblefor leptogenesis, flavour and CPviolation in charged-lepton
E
processes are different from those measurable in neutrino oscillation experiments, and also
P
motivate this paper.
0
As we discuss below, the predictions for these additional parameters vary in differ-
ent models. However, there are also several correlations. A combined analysis, using all 7
available processes, could give significant constraints on models, and even exclude certain (
classes of models. The joint study of phases, CP and lepton-flavour violation is a central
2
issue in this programme, and its better understanding is the central goal of this work.
0
The fact that the fermion mass matrices exhibit a hierarchical structure suggests that
0
they are generated by an underlying flavour symmetry for which the simplest possibility
would be an Abelian family symmetry. In such a case, one can parametrize the charges of 7
the Standard Model fields under the symmetry as in table 1. The Higgs charges are chosen )
so that the terms f fcH (where f stands for a generic fermion, the subscript 3 denotes the
3 3 0
thirdfamily,andH denoteseitherH orH )havezeronetchargewhereasothertermshave
1 2
5
non-zero charges. Then, in general only the (3,3) element of the associated mass matrix
2
will be non-zero as long as the U(1) symmetry remains unbroken. The remaining matrix
elements may be generated when the U(1) symmetry is spontaneously broken [13, 14] by
fields θ, θ¯that are singlets of the Standard Model gauge group, with U(1) charges that are
in most cases taken to be 1, respectively. Here we assume them to have equal vacuum
±
expectation values (vev’s). The suppression factor for each entry depends on the family
charge: the higher the net U(1) charge of a term f fcH, the higher the power n in the
i j
non-renormalisable term f fcH hθi n or hθ¯i n that has zero charge, where M is a
i j M M
high mass scale that is assumedht³o be´much³larg´erithan θ = θ¯ . The f fcH couplings
h i h i 3 3
are not suppressed, because they have no net charge.
Whilst such a family symmetry provides a promising origin for a hierarchical pattern
of fermion masses, in order to go further it is necessary to specify the charges of the
quarks, charged leptons and neutrinos [15]. In principle, one can fit the observed masses
and mixing angles by choices of flavour charges that are not constrained by additional
– 2 –
Q uc dc L ec νc H H
i i i i i i 2 1
U(1) α β γ c d e α β α γ
i i i i i i 3 3 3 3
− − − −
Table 1: Notation for possible U(1) charges of the various Standard Model fields, where i is a
generation index.
symmetries. However, the structure of the Standard Model is suggestive of an underlying
unification relating quark and lepton multiplets, as occurs in various GUT models. In such
a case, it is important to try to understand the different mass and mixing patterns and
othercorrelationspredictedbydifferentGUTs,andtheresultingpatternsofCPandlepton-
flavour violation in particular. This is the primary focus of this paper. Since particles in
the same multiplet of a GUT group have the same charge, the predictions and correlations
between observables willin general bedifferentfor different GUTs and charge assignments. J
A crucial question when studying flavour symmetries is whether these are indeed H
Abelian, as described above, or non-Abelian. In the case of an Abelian flavour symmetry,
E
takingpredictionsforthemassesandmixingsonly fromtheflavourandGUTstructureand
P
assuming that unknown numerical coefficients are generically (1) leads in the simplest
O 0
and apparently more natural neutrino models to large mass hierarchies and unacceptably
small solar mixing, which is correlated with the hierarchies of charged-lepton masses [16]. 7
Large solar mixing, as required by the experimental data, has therefore to arise either from (
the right-handed neutrino sector (as may arise, for instance, in see-saw models with domi-
2
nance by a single right-handed singlet neutrino, and other zero-determinant solutions), or
0
by imposing more U(1) symmetries and/or introducing more fields. However, this strat-
0
egy introduces additional model dependence and loses predictivity. On the other hand,
large atmospheric mixing is naturally predicted in a wide class of Abelian models, and 7
non-Abelian flavour symmetries typically predict naturally large angles for both solar and )
atmospheric neutrinos [17, 18].
0
Since there is no charge quantisation in Abelian groups, contrary to the non-Abelian
5
case, the symmetry cannot determine alone the numerical values of the mass-matrix ele-
2
ments and hence the mixing angles and phases. Nevertheless, information on the phases
and values of the coefficients can be inferred phenomenologically, and individual Abelian
models may be used to make interesting qualitative predictions, just as they have for the
mixing angles, based on the powers of the small parameters θ /M, θ¯ /M that are allowed
h i h i
by the symmetries. However, any predictivity requires the existence of stable solutions
where a small change in the parameters does not change drastically the predictions. Non-
Abelian symmetries are more predictive in this respect. However, once the symmetry is
broken (as is required to generate large lepton mass hierarchies) this predictivity is to a
certain extent lost. Even in this case, therefore, the phases are derived by a combination
of theoretical and phenomenological considerations, again in a model-dependent way.
The fact that neutrinos have masses and mix with each other like quarks also im-
plies that, in principle, we can expect non-negligible CP-violation in the neutrino sector.
This may manifest itself in several different ways in low-energy neutrino physics, including
– 3 –
the Maki-Nakagawa-Sakata (MNS) oscillation phase δ and two possible Majorana phases.
Additional phases arise in extensions of the light-neutrino sector to include either heavy
singlet neutrinos and/or charged leptons. For example, even the minimal three-generation
see-saw model has 6 independent CP-violating phases, and leptogenesis is independent of
the light-neutrino phases, in particular. On the other hand, some of the phases on which
leptogenesis does depend may in principle be observable in the charged-lepton sector, in
the presence of low-energy supersymmetry. As we discuss below, these observables may
take very different values in classes of models that fit the present neutrino data equally
well.
The structure of this paper is as follows. In section 2 we discuss the neutrino and
charged-lepton observables for which we investigate the predictions of various GUTs with
Abelian flavour symmetries. The specific GUT models and their predictions are discussed
in section 3. Our conclusions are set out in section 4, and appendices present details of the
J
specific GUTs studied.
H
E
2. Observables in the lepton sector
P
2.1 Neutrino observables
0
The mixing matrices V ,U that yield physical CP and flavour violating observables
e,D R,ν 7
are those making the following diagonalisations:
(
VTY Y†V∗ = Diag(y2,y2,y2), (2.1) 2
e e e e e µ τ
VTY Y†V∗ = Diag(y2,y2,y2), (2.2) 0
D ν ν D ν ν ν
UTM U = Diag(M ,M ,M ), (2.3) 0
R RR R 1 2 3
UTm U = Diag(m ,m ,m ), (2.4) 7
ν eff ν ν1 ν2 ν3
)
where Y stand for the Yukawa couplings of neutrinos and charged leptons respectively,
e,ν
0
and M is the heavy Majorana neutrino mass matrix, which we assume to be three-
RR
dimensional (for a review see [19]). The effective neutrino mass matrix is: 5
2
1
m mν mν T. (2.5)
eff ≈ DM D
RR
In terms of the above matrices, the Maki-Nakagawa-Sakata (MNS) matrix becomes
U U = V U†, (2.6)
MNS ≡ e ν
and can be parametrized as:
U = diag(eiδe,eiδµ,eiδτ).V.diag(e−iφ1/2,e−iφ2/2,1), (2.7)
where
c c s c s e−iδ
12 13 12 13 13
V = c s s s c eiδ c c s s s eiδ s c , (2.8)
 23 12 23 13 12 23 12 23 13 12 23 13 
− − −
s s c s c eiδ s c c s s eiδ c c
23 12 23 13 12 23 12 23 13 12 23 13
 − − − 
 
– 4 –
and c and s stand for cosθ and sinθ , respectively. In this formalism, the three
ij ij ij ij
neutrino mixing angles and the six CP-violating phases are given by
U U
12 23
θ = arcsin(U ), θ = arctan | | , θ = arctan | | ,
13 13 12 23
| | U U
µ| 11|¶ µ| 33|¶
Ui∗iUijUjiUj∗j +c c s
δ = arg(U ), δ = arg(U ), δ = arg c12c213c23s13 12 23 13 ,
µ 23 τ 33
−  s s 
12 23
δ = arg(eiδU ), φ = 2arg(eiδeU∗ ), φ = 2arg(eiδeU∗ ) , 
e 13 1 11 2 12
where i,j = 1,2,3 and i = j. A useful measure of the amount of CP-violation in the
6
oscillations of light neutrinos is the Jarlskog invariant
1
J = Im(U∗ U U U∗ ) J
CP 2| 11 12 21 22 |
1 H
= Im(U∗ U U U∗ )
2| 11 13 31 33 | E
1
= Im(U∗ U U U∗ ) P
2| 22 23 32 33 |
1 0
= c c2 c sinδs s s . (2.9)
2| 12 13 23 12 13 23| 7
It is clear that different models that reproduce the experimental values of θ and (
12,23
the differences in light neutrino masses-squared may well predict different amounts of CP- 2
violation in neutrino oscillations (2.9) and elsewhere. So far, experiment has provided only
0
an upperboundonthepossiblemagnitudeofθ , andnoinformation onδ. Themagnitude
13
0
of CP-violation in neutrino oscillations may even be zero, for example in models with
texture zeroes in the (1,3) entries. As we see later, specific models may connect θ and 7
13
δ, and also provide connections to other observables. We aim here at the differentiation of )
possible flavour models according to their predictions for observables in the neutrinosector
0
- what are the expected magnitudes of θ ,δ and φ ? - and elsewhere.1 We now discuss
13 2,3 5
some of the other observables in more detail.
2
2.2 Leptogenesis
The textures of neutrino Yukawa couplings may be constrained by the requirement of suc-
cessful leptogenesis [21], which requires obtaining the correct magnitude of lepton-flavour
violation (LFV) in the decays of heavy, right-handed Majorana neutrinos via a difference
between the branching ratios (BR) for heavy-neutrino decays into leptons and antileptons:
BR(Nc Φ+ℓ) = BR(Nc Φ+ℓ).
L → 6 L →
Since non-perturbative lepton- and baryon-number-violating interactions mediated by
sphalerons are in thermal equilibrium up to the time of the electroweak phase transition,
1In order to calculate mixing angles and CP-violating phases in an automated way, we analyze the
different models using thepackage REAP [20].
– 5 –
a non-zero lepton number gives rise to a non-zero baryon number, by sharing the lepton
asymmetry ∆L = 0 with a baryon asymmetry ∆B = 0.
6 6
Inordertoavoidwashoutoftheinitialdecay asymmetrybyperturbativedecay, inverse
decay and scattering interactions, the model must satisfy an out-of-equilibrium condition,
namely that the heavy-neutrino decay rate is smaller than the Hubble parameter H at
temperatures of the order of the right handed neutrino masses. The tree-level width of the
heavy neutrino N with mass M is: Γ = [(λ†λ) /8π]M , which should be compared with
i i ii i
the Hubble expansion rate H 1.7 g1/2 T2/M , (where gMSSM 228.75, gSM = 106.75
≈ ∗ P ∗ ≈ ∗
and M is the Planck mass), leading to the requirement
P
(λ†λ)
ii
M < M .
P i
1/2
14πg
∗
J
Here, λ Y′ denotes the neutrino Yukawa matrix in the basis where the Majorana masses
≡ ν
M are diagonal. This condition may be implemented more accurately by looking in detail H
i
at the Boltzmann equations, but this formula is sufficient for our purposes. E
TheCP-violating decay asymmetry ǫj in the decay of a heavy-neutrino flavour j arises P
from the interference between tree-level and one-loop amplitudes, and is
0
1 M2 7
ǫ = Im ((Y′†Y′) )2 f j , (2.10)
j (8πYν′†Yν′)jj Xi6=j h ν ν ji i ÃMi2! (
2
where the kinematic function
0
1 1+y 0
f(y) = √y +1 (1+y)ln .
1 y − y 7
· − µ ¶¸
)
The first term in f(y) arises from self-energy corrections, while the second and third terms
0
arise from the one-loop vertex. For models with degenerate Majorana masses, one expects
a resonant enhancement of the lepton asymmetry, since in this case y 1 and 5
∼
2
M
i
f(y) | | . (2.11)
∼ −2(M M )
j i
| |−| |
This will be manifest in some of the examples below.
The lepton asymmetry produced by the decays of the heavy neutrinos is in general
diluted partially by lepton-number-violating processes [21]. The washout factor κ is ap-
i
proximately
10−3eV m˜ −0.6
i
κ (m˜ ) 0.3 log , (2.12)
i i ≃ m˜ 10−3eV
µ i ¶µ ¶
where
m˜ = (M′† M′ ) /M , (2.13)
i RR RR ii | i|
and the primes denote the basis where the right-handed neutrino mass matrix is diagonal.
– 6 –
Then, for a supersymmetric model, the lepton and baryon asymmetries are given by
n 1
L 0.8 10−2κ ǫ , (2.14)
i i
s ≃ × ∆
³n ´ 1
B 2.8 10−3κ ǫ , (2.15)
i i
s ≃ − × ∆
³ ´ 1
η 0.02κ ǫ , (2.16)
B i i
≃ − ∆
wheretheentropyoftheco-movingvolumesisgivenbys = (2/45)g π2T3,andg = 228.75
∗ ∗
refers to the effective number of relativistic degrees of freedom contributing to the entropy
in our supersymmetric model.
In the above relations, ∆ is a dilution factor that appears due to entropy production
from symmetry breaking and an inflationary phase. This effect has been studied in [22],
in the context of the breaking of SU(5) U(1) when a singlet field Φ gets a vev. In this
case, the dilution factor is [23] ∆ = s(RdΦ×)(RdΦ)3 V3m3η/2 . Here, the Φ decay rate J
s(Rdη) Rdη ∼ αΦ1/2 m3S/U2SYMP3 H
is given by ΓΦ = αΦm3SVU2SY, V is the scale where the vev of the Higgs 10 and 10 break the E
flipped-SU(5) group, m is the inflaton mass, and m is the supersymmetry breaking
η SUSY P
scale.
0
Successful leptogenesis would require [25]:
7
η = (6.15 0.25) 10−10. (2.17)
B
± × (
Clearly, κi/∆ < 1 in general and a model with too small a value of ǫi would not be valid. 2
However, as we discuss below, this is not the case in the models we study, which generally
0
require ∆ 1 in order to provide a successful scheme of leptogenesis. An alternative
≫ 0
to invoking a large dilution factor would be to adopt a wider range of neutrino Yukawa
couplings and Majorana masses that can lead to consistent solutions. 7
The impact of flavour effects in leptogenesis has also been considered in [26] and more )
recently in [27], and may affect the results by factors of 2. Since the conclusions of our
0
∼
paper are not affected by the possible presence of such factors, we will not proceed with
5
more detailed considerations in this direction.
2
2.3 Charged-lepton-flavour violation
In the context of low-energy supersymmetry, mixing in the neutrino sector also gener-
ates mixing in the sleptons via loop corrections, contributing to rare decays and lepton-
flavour conversions. We evaluate these effects in the context of the CMSSM, where the
soft supersymmetry-breaking masses of the charged and neutral sleptons are assumed to
be universal at the GUT scale, and may be written in diagonal form with a common value
m . Off-diagonal entries in the slepton mass matrix m2 are then generated radiatively by
0 L˜
the renormalization-group evolution from the GUT scale M .
GUT
The branching ratios for LFV decays can be described well by a single-mass-insertion
approximation [28, 29]:
α3
BR(l l γ) (m ,M ,µ)m2 2tan2β, (2.18)
i → j ≈ G2 F 0 1/2 | L˜ij|
F
– 7 –
where is a function of the soft supersymmetry-breaking masses fixed at the high scale
F
M , and i= j are generation indices. In the commonly-used approximation for solving
GUT
6
the renormalisation-group equations of a single intermediate right-handed neutrino thresh-
old, the mass corrections are related to the neutrino Yukawa couplings by [30]
m2 = κ Y¯ ik(∆t+∆ℓ )(Y¯jk)∗, (2.19)
L˜ij ν k ν
k
X
where Y¯ is the Dirac neutrino Yukawa matrix in a basis where both the heavy-Majorana
ν
and charged-lepton couplings become diagonal, and
κ = 6m2 2A2, ∆t = ln(M /M )/16π2, ∆ℓ = ln(M /M )/16π2, (2.20)
− 0− 0 GUT 3 k 3 k
where M with i= 1,2,3 denote the Majorana neutrino masses assuming M < M < M .
i 1 2 3
Thisresultisnotveryaccurate,butitisausefulapproximationforobtaininganalytical J
expressions for lepton-flavour-violating decay rates [31]. In the analysis that follows we
H
assume for simplicity that A = 0. If A = 0, this parameter would also be similarly
0 0
6 E
renormalized, via
1 M P
(δA ) A Y (Y¯ Y¯†) log GUT. (2.21)
e ij ≈ −8π2 0 ei ν ν ij M
i 0
The extra suppression due to Y and the form of κ indicate that, unless A becomes much
e 0 7
larger than m , our conclusions will be qualitatively unchanged.
0
(
Another lepton-number-violating observable is µ e conversion on a nucleus, which
→
2
has a rate
0
α E p ZF2
R(µ+Ti e+Ti) e e c BR(µ eγ)
→ ≈ 3π m2 Cf(A,Z) → 0
µ
5.6 10−3BR(µ eγ). (2.22) 7
≈ × →
)
This process is very interesting, despite the relative suppression by about two orders of
0
magnitude,becauseoftheaccuracypossibleinfuturemeasurementsofthisprocess. Within
the framework discussed here, a similar suppression is expected for the decay µ 3e, 5
→
2
Γ(µ+ e+e+e−)
→ 6 10−3. (2.23)
Γ(µ+ e+γ) ≈ ×
→
However, the present experimental bound on this branching ratio is relatively weak, and
the prospects for significant improvement are more distant. It should be noted, though,
thatthisdecay doesofferthepossibilityofobservingaT-violatingasymmetry,whichmight
be observable if µ eγ occurs with a rate close to the present experimental upper limit.
→
The flavour-violating decays τ eγ and µγ are also potentially interesting. They are
→
governed by formulae similar to those above, and later we present some results for them.
3. GUTs with abelian flavour symmetries
As mentioned in the Introduction, we focus in this work on Abelian flavour symmetries,
sincetheyaresimpleandarisenaturallyinawideclassofmodels. Moreover, understanding
– 8 –
Abelian symmetries is a first step towards the understanding of non-Abelian symmetries.
Indeed, bycombiningtwo ormoreAbelian symmetriesandintroducingmorethan onefield
whose vev determines the expansion parameter of the mass matrices, one can simulate to
some extent the picture that one would obtain from a non-Abelian structure.
3.1 SU(5)
The first GUT group we analyze is SU(5), whose minimal matter field content is three
families of (Q,uc,ec) 10 representations of SU(5), three families of (L,dc) 5 repre-
i i
∈ ∈
sentations, and heavy right-handed neutrinos Nc ν in singlet representations. Only
L ≡ R
two heavy Nc ν fields are needed to provide two non-zero masses for light neutrinos,
L ≡ R
as required by experiment, and more than three are present in some models. However, in
what follows, we assume that there are also just three heavy Nc ν fields. This model
L ≡ R
has the following properties that are important for our analysis:
J
(i) the up-quark mass matrix is symmetric, H
E
(ii) the charged-lepton mass matrix is the transpose of the down-quark mass matrix,
P
which relates the mixing of the left-handed leptons to that of the right-handed down-
typequarks. SincetheCKMmixinginthequarksectorisduetoamismatchbetween 0
the mixing of the left-handed up- and down-type quarks, it is independent of mixing 7
in the lepton sector. In particular, in SU(5) the large mixing angle that is observed
(
in atmospheric neutrino oscillations can easily be consistent with the observed small
2
V mixing. Most importantly,
CKM
0
(iii) it is clear from theSU(5) representation structurethat the Abelian flavour charges of
0
the fermions (Q,uc,ec) in the same 10 representation must be identical, as must the
i
7
charges of the (L,dc) in the same 5 representation. This is the type of correlation
i
that we seek to test by examining model predictions for flavour and CP violation in )
the generalised lepton sector. 0
5
There are several possibilities for the Abelian flavour charges, that are motivated by
theoretical considerations such as anomaly cancellation [32] as well as phenomenological 2
arguments. Within this framework, one may address questions such as:
How close to maximal is the atmospheric mixing?
•
How large is the solar mixing?
•
How large are θ and δ?
13
•
At which level is mixing controlled by the hierarchies of charged-lepton masses, and
•
how strong is the influence from the heavy right-handed neutrino sector?
What statements can be made about other observables?
•
Thereareseveral viable sets of textures that weanalyze foranswers tothese questions.
Taking into account the above-mentioned constraints arising from the SU(5) multiplet
– 9 –