Neutrino Masses and Particle Physics Beyond the Standard … · Neutrino Masses and Particle Physics Beyond the Standard Model Hugo Duarte de Jesus Alves Teixeira Serôdio Dissertação

Neutrino Massesand

Particle Physics Beyond the Standard Model

Hugo Duarte de Jesus Alves Teixeira Serôdio

Dissertação para obtenção do Grau de Mestre em

Engenharia Física Tecnológica

JúriPresidente: Prof. Gustavo Castelo Branco

Orientador: Prof. Gustavo Castelo Branco

Vogais: Prof. Jorge Crispim Romão

Doutor David Emmanuel-Costa

September 2007

Aos meus Pais e Irmão,

“In a gauge invariant theory a localquantity with vanishing mean valueon its orbit under the action of thegauge group has zero state expectationvalue.”

Elitzur’s Theorem

Acknowledgements

During these last two years, many people have contributed to the development of my work. First ofall, I would like to express my sincere gratitude to all members of CFTP (Centro de Física Teórica dePartículas) for the excellent working conditions and stimulating environment. In particular, I thankthe CFTP secretaries, Sandra Oliveira and Fátima Casquilho for their kindness and assistance.

I am indebted to Fundação para a Ciência e a Tecnologia for the undergraduated scolarships overthe last two years and to CFTP for the important financial support which allowed me to participatein some international scientific events.

I express my enormous gratitude to Doctor Ricardo González Felipe for his help in many aspectsof this thesis and specially to Doctor David Emmanuel-Costa for his patience and time wasted onme.

To my colleagues and friends Ana Roque, Catarina Simões, Carlos Afonço, Filipe Marques, JoãoLaia, Marco Cardoso, Miguel Costa Lopes, I am grateful for all our good moments together andstimulating discussions within and outside physics. A special thank to Francisca Figueiredo for hersupport and English grammar orientation.

I wish to devote a special acknowledgement to my advisor, Professor Gustavo Castelo-Branco,who has always supported me and gave me the opportunity to work with excellent researchers.

Finally, my gratitude goes to my parents for their support and love, and a special thank to mybrother, for the uncountable nights that he picked me up after midnight at the university.

vii

Resumo

Compreender a origem das massas dos neutrinos e as suas implicações fenomenológicas é um dostemas mais estimulantes na física de partículas moderna. Dar massas aos neutrinos é apenas possívelno contexto de física para além do modelo padrão das interacções electrofracas. Uma explicaçãosimples e apelativa para o valor tão pequeno das massa dos neutrinos é fornecido pelo mecanismode seesaw que pode ser fácilmente implementado como uma extensão miníma do modelo padrão ouatravés de teórias mais elaboradas.

Para o estudo dos neutrinos é necessário um conhecimento de base da física do modelo padrãodas patículas elementares. Em particular, o estudo de teorias de gauge, mecanismo de Higgs, teoriaelectrofraca e extenções, foi necessário de maneira a obter o conhecimento base para entrar na áreados neutrinos.

Esta tese está dividida em duas partes; uma relacionada com a física básica necessária para aconstrução do modelo padrão das interacções electrofracas, e uma segunda parte onde se focaramextensões mínimas necessárias para incluir a massa dos neutrinos, bem como oscilações e texturas demassa para os neutrinos, com especial ênfase neste assunto.

Palavras-chave:Extensões ao Modelo Padrão; Massa dos Neutrinos; Mecanismo de Seesaw; Mistura Leptónica; Mod-elo Padrão

ix

Abstract

Understanding the origin of neutrino masses and its phenomenological implications is one of the mostchallenging subjects of modern particle physics. Massive neutrinos can only be realized in the lightof new physics beyond the standard model of weak interactions. A simple and appealing explanationfor the smallness of neutrino masses is provided by the seesaw which can be easily implemented in aminimal extension of the standard model or in other more elaborated theories.

To study neutrinos it is needed a background in the standard model of particle physics. Therefore,a study of gauge theories, Higgs mechanism, electroweak theory and extensions was made, in orderto obtain the basic knowledge needed to proceed into the neutrino field.

This thesis is divided into two parts; one related to the basics tools needed in order to constructthe standard model of weak interactions, and a second part that focus the minimal extensions neededto include neutrino masses, as well as oscillation and mass schemes for neutrinos are study, withspecial emphasis in the last one.

Key-words:Extensions of the Standard Model; Leptonic Mixing; Neutrino Masses; Seesaw Mechanism; StandardModel extensions.

xi

Contents

Resumo ix

Abstract xi

List of Tables xv

List of Figures xviii

Abbreviations xxi

Preface 2

I The Standard Model 3

1 Gauge Theories 5

1.1 Gauge invariance in Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Global Abelian U(1) and non-Abelian SU(2) symmetry . . . . . . . . . . . . . 7

1.2 Local Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Abelian U(1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Non-Abelian SU(2) gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Spontaneous Symmetry Breaking and Higgs mechanism 13

2.1 Discrete symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Continuous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Standard Model construction 19

3.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Electroweak symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Charged and Neutral currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Fermion masses and mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Quark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Leptonic sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Accidental symmetries of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Final remarks of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

xiii

II Beyond the Standard Model 39

4 Massive Neutrinos in minimal Standard Model extensions 414.1 Dirac Vs. Majorana mass term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 The New-Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 The seesaw mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Leptonic mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Neutrino Oscillations 515.1 Oscillation in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1.1 Two flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.2 Three flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Oscillation in matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.1 Evolution equation. Two flavour case. . . . . . . . . . . . . . . . . . . . . . . . 57

5.3 Experimental neutrino physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.1 Solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.2 Atmospheric neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Double beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Texture zeros of the fermion mass matrices 676.1 Four zero schemes of quark mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 Lepton texture zero mass matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2.1 The (1,1) and off diagonal zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2.2 Four-zero textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.3 Phenomenological implications of the four texture zero Ansätze . . . . . . . . . 74

7 Concluding remarks 83

A Chirality Vs. helicity 85

B CP, T and CPT symmetries in neutrino oscillations. 87

xiv

List of Tables

1.1 Symmetries and conserved quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Action and equation of motion for fields used in this thesis. We use the units where

c = ~ = 1. Therefore,[x−1

]= [m] = [E] = eV . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 The Standard Model parameters. The u, d and s quark masses are estimates of so-called ‘current-quark masses’, in a mass-independent subtraction scheme such as MSat scale µ ' 2GeV. The c and b quark masses are the ‘running’ masses in the MSscheme. The top quark mass is given by direct observations of top events.[6] . . . . . . 37

5.1 Oscillation experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Solar neutrino experimental results. The Homestake, GALLEX, SAGE and GNO

data are given in SNU(1SNU= 10−36 events/atom/second), the Kamiokande, Super-Kamiokande and SNO data are measured in 106cm−2s−1.[2] . . . . . . . . . . . . . . . 62

5.3 muon-electron ratio for several experiments.[34] . . . . . . . . . . . . . . . . . . . . . . 62

6.1 Neutrino mass matrix patterns, which are in accord with the current experimentaldata.[26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Best-fit values, 2σ, 3σ and 4σ intervals (1 d.o.f.) for the three-flavour neutrinos oscil-lation parameters from global data including solar, atmospheric, reactor (KamLANDand CHOOZ) and (K2K) experiments.[31] . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3 Ansatz Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xv

List of Figures

2.1 The two possible potentials regarding the sign of µ2. . . . . . . . . . . . . . . . . . . . 142.2 This represents the shape of the potential when µ < 0. The real part Re(φ) is φ1 and

Im(φ) is φ2. This can also be interpreted in terms of η and χ, where η is referred tothe radial oscillations and χ to the angular oscillations. . . . . . . . . . . . . . . . . . 16

3.1 Fermi theory for the neutron decay.[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 IVB theory for the neutron decay.[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Interactions between the gauge bosons Z0 and W± with the Higgs boson.[1] . . . . . 253.4 Charged current interaction vertices.[1] . . . . . . . . . . . . . . . . . . . . . . . . . . 263.5 Neutral current interaction vertices.[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 Self interaction of the gauge bosonsW±, Z0 and γ. Notice that there is no vertex with

only Z0, γ our both because they do not have any charge.[1] . . . . . . . . . . . . . . . 283.7 Interaction vertex between the Higgs boson and the fermions.[1] . . . . . . . . . . . . . 293.8 Flavour changing transitions through the charged current coupling with theW± bosons.

In this figure one easily sees that the interaction vertex in charged current processesis proportional to the element of the mixing matrix. In the second figure one has theflavour changing amplitude.[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.1 Neutrino oscillations in vacuum.[30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Neutrino mass spectrums.[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3 Neutrino interactions with matter.[30] . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4 Energy generation in the Sun via the pp chains.[2] . . . . . . . . . . . . . . . . . . . . 605.5 The predicted unoscillated spectrum dΦ/dEν of solar neutrinos, together with the

energy thresholds of the experiments performed so far and with the best-fit oscillationsurvival probability Pee(Eν) (dashed line).[2] . . . . . . . . . . . . . . . . . . . . . . . 61

5.6 The main Super-Kamiokande data: number of e± (red) and of µ± (blue) events asfunctions of direction of scattered lepton. The horizontal axis is cos θ, the cosine of thezenith angle ranging between -1 (vertically up-going events) and +1 (vertically down-going events). The third plot shows high-energy through-going muon, only measuredin the up direction. The crosses are the data and their errors, the thin lines arethe best-fit oscillation expectation, and thick lines are the no-oscilation expectation:these are roughly up/down symmetric. Data in the multi-GeV muon samples are veryclear asymmetric, while data in the electron samples (in red) are compatible with nooscillation. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.7 (a) two-neutrino double beta decay. (b) neutrinoless double beta decay . . . . . . . . 645.8 Region allowed for the |(Mν)ee|.[7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

xvii

6.1 In this Ansatz with normal hierarchy all the experimental boundaries are verified.In figure a) is represented the plot of |Ue3|Vs.m1 and in figure b) it is plotted the|(Mν)ee|Vs.m1, which is a physical quantity that can be measured in the lab. . . . . . 77

6.2 In a) is represented the |Ue3|Vs.m3, showing that only the region with high values forthe mass of m3 is allowed. In b) is plotted sin2 θ12Vs.|Ue3|, show no region allowedfor this Ansatz. Therefore, these plots excludes the Ansatz of Class I with invertedhierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.3 In a) it is plotted |Ue3|Vs.m1, where we can see a lower bound for |Ue3|. In b) it isrepresented the |(Mν)ee|. Both plots correspond to the normal hierarchy case. . . . . . 80

6.4 Plot of the |(Mν)ee| Vs. m3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.5 In a) is represented the sin2 θ23Vs.m1 and in b) is represented |(Mν)ee| Vs. m1. . . . . 816.6 In a) is represented the sin2 θ12Vs.|Ue3| and in b) is represented |(Mν)ee| Vs. m3. . . . 81

xviii

Abbreviations

CC Charged Current

CKM Cabbibo-Kobayashi-Maskawa

CP Charge-Parity

FCNC Flavour Changing Neutral Current

FGM Frampton Glashow Marfatia

GALLEX Gallium Experiment

GIM Glashow-Iliopoulos-Maiani

GUT Grand Unified Theory

IMB Irvine Michigan Brookhaven

INMS Inverted Neutrino Mass Spectrum

IVB Intermediate Vector Bosons

KamLAND Kamioka Liquid scintillator Anti-Neutrino Detector

K2K from KEK to Kamioka

LEP Large Electron-Positron Collider

LGI Local Gauge Invariance/Invariant

LSND Liquid Scintillator Neutrino Detector

MACRO Monopole, Astrophysics and Cosmic Ray Observatory

MiniBooNE Mini Booster Neutrino Experiment

MINOS Main Injector Neutrino Oscillation Search

NC Neutral Current

NNMS Normal Neutrino Mass Spectrum

PDG Particle Data Group

PMNS Pontecorvo-Maki-Nakagawa-Sakata

QCD Quantum Cromodynamics

xx

QED Quantum Electrodynamics

SAGE Soviet-American Gallium Experiment

SM Standard Model

SNO Sudbury Neutrino Observatory

SSB Spontaneous Symmetry Breaking

SSM Standard Solar Model

νSM New Standard Model

WB Weak Basis

WBT Weak Basis Tranformation

xxi

Preface

For a long time, the standard model of weak interactions seemed to be satisfactory in the explana-tion of low-energy phenomena. Apart from some fundamental theoretical questions, the Glashow-Weinberg-Salam theory was in a very good agreement with experiment until recently in the time scaleof particle physics. Still, even to an outsider the standard seems to be somehow incomplete. Onebecomes aware of this when we try to explain the very simple features of the standard model. Onestart introducing the particle content, six massive quarks, three massive charged leptnos and threeother particles called neutrinos that are massless. At this point several questions arise: Why are theneutrinos massless?, Why do the particles have such a hierarchy in the masses?, Are these particlesall connected somehow?

These questions are not answered by the standard model, which is the same as saying that thestandard model no longer completely describes the low-energy world.

Understanding the origin of neutrino masses is nowadays one of the most puzzling topics of particlephysics. What truly sets it apart from other equally enticing subjects is that, up to now, the onlyevidence of a non-standard physical phenomenon requires the neutrinos to be massive and mixed.Experimental results provided by neutrino oscillation experiments constitute the first solid evidencein favour of the existence of physics beyond the standard model.

As is well known since the early beta decay experiments, neutrinos are extremely light whencompared with the other fermions. This is hard to understand in the framework of a minimalextension of the standard model where neutrinos become massive via the same mechanism as chargedleptons and quarks do. In other words, it seems quite unnatural that neutrinos have Dirac masses inthe sense that this drives us into another hierarchy problem of the theory. Therefore, one is lead tospeculate about the existence of some dynamical mechanism responsible for the smallness of neutrinosmasses. Among all the scenarios proposed in the last years, the seesaw mechanism appears to be themost simple and natural one. Its beauty resides not only on the mass suppression by itself, but alsoon the fact that neutrino mass eigenstates are predicted to be Majorana particles. Remarkably, thismay be telling us that such tiny neutrino masses are most likely linked to their Majorana nature.One can now wonder about how difficult implementing the seesaw mechanism is on the grounds ofan underlying theory. Again, the seesaw scores since it can be realized in almost any theoreticalframework, provided that heavy right handed neutrinos are included.

The last three decades have been extremely exciting for neutrino physics. A major break-through occurred in 1998 with the evidence in favour of neutrino oscillations reported by the Super-Kamiokande experiment. Nowadays, several neutrino experiments are running and others beingproposed or under construction with the ultimate goal of revealing the properties of neutrinos. Astrategy has been planned in such a way that neutrinos are being ‘attacked’ from all fronts. The

1

currently available and future results are and will be of extreme importance and some of them willsurely be decisive.

The first part of this thesis, divided in three chapters, is devoted to some relevant tools neededfor the understanding of neutrino physics. We start with a brief review of gauge theories, global andlocal. There we present the importance of symmetries in physics, the Noether and Nambo-Goldstonetheorems, and also introduce the concept of covariant derivative. We then present the buildingblocks of the spontaneous symmetry breaking and Higgs mechanism. In Chapter 3 we construct thestandard model of particle physics. We start by a little historical point of view, introducing the Fermifour-point interaction theory as an effective theory, that at high energies would have its origins inan intermediate vector boson theory. One start then writhing the standard model Lagrangian andspontaneously breaking the electroweak part. We introduce the charged and neutral currents, thefermion masses and mixings, as well as a brief note about accidental symmetries in this model.

The second part is completely dedicated to neutrinos. In chapter 4 we present several possibilitiesto introduce massive neutrinos into the standard model. We talk about the possible Dirac or Majorananature of the neutrinos and then introduce the seesaw mechanism and finish the chapter with thestudy of the leptonic mixing. Chapter 5 is dedicated to neutrino oscillations in vacuum and matter,the double beta decay and neutrinoless double beta decay are also mentioned. Chapter 6 is devotedto the study of zero textures and mass matrices schemes, in the leptonic sector. In order to do thatwe first introduce the quark sector, such that the extension to the leptonic sector is done a naturalconsequence. We study some physical implications of the mass schemes, specially the ones relatedwith the neutrinoless double beta decay.

Finally, in the Conclusions, we outline the most relevant aspects discussed in this thesis.

2

Part I

The Standard Model

3

Chapter 1

Gauge Theories

Contents1.1 Gauge invariance in Field Theory . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Global Abelian U(1) and non-Abelian SU(2) symmetry . . . . . . . . . . . 7

1.2 Local Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Abelian U(1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Non-Abelian SU(2) gauge symmetry . . . . . . . . . . . . . . . . . . . . . . 10

Symmetries have played a major rôle in the development of theoretical physics. In fact, fromClassical Mechanics to Particle Physics, most models have been derived from the principle that thereare symmetries ruling the physical world. For example, suppose that one watches some physicalevent in a mirror. Does the event in the mirror correspond to something allowed by the laws ofNature? From this question rises the concept of P symmetry (or Parity), consisting in the invarianceof systems under a discrete transformation, where the sign of the coordinates x, y and z changes.Will the events seen in a backward-running film look possible and realistic, or will they be at oddswith the laws of Nature? This is the issue raised by the T symmetry (or Time reversal), consistingin the invariance of systems under a discrete symmetry, where the sign of the coordinate t changes.Contrary to P and T , the C symmetry (or charge-conjugation) does not have an analogue in classicalphysics. This symmetry is related to the existence of an antiparticle for every particle. C symmetryasserts that antiparticles behave in exactly the same way ast the corresponding particles.

These are some of the so-called discrete symmetries, in opposition to them we have the continuousones. Some examples of these are the invariance under a translation or a rotation, which gives riseto the conservation of linear and angular moment respectively. More detailed information aboutsymmetries can be found in[3].

In this chapter we will introduce the concepts of global and local gauge symmetries for the caseof Abelian and non-Abelian groups.

1.1 Gauge invariance in Field Theory

Conservations laws in physics can be attributed to symmetry principles. The invariance of a physicalsystem under certain symmetry transformations implies an appropriate set of conservation laws. Thisis the Noether’s theorem which states:

5

To each continuous symmetry there is a corresponding conservation law, a conserved cur-rent, and vice versa.

One can see some examples in classical and quantum mechanics. We recall that in quantummechanics observables are associated with operators and their evolution in time is given by dO

dt =

i [H,O] (Heisenberg picture).

Symmetries Classical mechanics Quantum mechanics Conserved quantities

t→ t+ a dEdt =0 [H,E] = 0 Energy

ri → ri + bidpidt = 0 [H, pi] = 0 Linear momentum

ri → RijrjdJidt = 0 [H,Ji] = 0 Angular momentum

Table 1.1: Symmetries and conserved quantities.

Particle physics is formulated using the language of quantum field theory. A field is a generalizationof the motion of a generalized coordinate, not just to several particles, but to a continuum of particles,with the possibility of one or more at each point in space. The basic object is the action

S =∫d4xL(Φi(x), ∂µΦi(x)) , (1.1)

where Φ(x) denotes a bosonic or fermionic field, i.e., Φ(x) ∈ {φ(x), ψ(x)}. Let us consider a La-grangian density, L, invariant under a symmetry based on a particular Lie group, G, 1 in which thefields transform infinitesimally as

Φi(x)→ Φ′i(x) = Φi(x) + δΦi(x) , where δΦi(x) = iεataijΦj(x) , (1.2)

εa are (x-independent) small parameters and the ta are set of matrices satisfying the Lie algebra[ta, tb

]= iCabctc , (1.3)

where the Cabc are the structure constants, characteristic of the Lie group G. The transformation inEq.(1.2) corresponds to the following change in the Lagrangian density

δL =δLδΦi

δΦi +δL

δ(∂µΦi)δ(∂µΦi) . (1.4)

Using the equation of motion

∂µδL

δ(∂µΦi)− δLδΦi

= 0 (1.5)

and the fact that δ(∂µΦi) ≡ ∂µΦ′i − ∂µΦi = ∂µ(δΦi), we can write δL as

δL = εa∂µ

[δL

δ(∂µΦi)itaijΦj

]. (1.6)

Since the Lagrangian is invariant under Eq.(1.2), i.e. δL = 0, Eq.(1.6) implies a conserved current

∂µJaµ = 0 , with Jaµ = −i δLδ(∂µΦi)

taijΦj . (1.7)

We can then define the conserved charges by

Qa =∫d3xJa0 (x) , (1.8)

1The Lie groups play an essential rôle in many aspects of particle physics.

6

which are the generators 2 of the symmetry group. The number of charges is equal to the numberof generators and they satisfy the commutation relations[

Qa, Qb]

= iCabcQc . (1.9)

This result is another example of the Noether’s theorem, this time applied to field theory. In table1.2 we present the type of fields that we will be working with, their respective action and equation ofmotion. In subsection 1.1.1, we will continue with the study of global symmetries for the particularcases of Abelian and non-Abelian groups with fermionic fields. In section 1.2 we will extend thatstudy for the case of Local Gauge Invariance (LGI).

Fields Action Equation of motion

Real scalar, ϕ[ϕ] = E

∫d4x

(∂µϕ∂

µϕ− 12m

2ϕ2)

(�2 +m2)ϕ = 0

Complex scalar, φ[φ] = E

∫d4x

(∂µφ∂

µφ∗ −m2 | φ |2) (�2 +m2)φ = 0

(�2 +m2)φ∗ = 0

Spin 1/2, ψ[ψ] = E3/2

∫d4x ψ (iγµ∂µ −m)ψ

(iγµ∂µ −m)ψ = 0ψ(iγµ∂µ +m) = 0

Massless vector, Aµ[Aµ] = E

∫d4x

(− 1

4FµνFµν − jµAµ − 1

2ξ (∂µAµ))

∂µFµν + 1

ξ∂ν(∂µAµ) = jµ

Table 1.2: Action and equation of motion for fields used in this thesis. We use the units wherec = ~ = 1. Therefore,

[x−1

]= [m] = [E] = eV .

1.1.1 Global Abelian U(1) and non-Abelian SU(2) symmetry

Symmetries which are characterized by space-time independent parameter in the infinitesimal trans-formations of the fields are called global symmetries. We consider the Lagrangian density for aspin-1/2 free field (similar analysis can be also obtain for spinless boson fields)

LDirac = ψ(x)(iγµ∂µ −m)ψ(x) , (1.10)

and perform a U(1) infinitesimal transformation

ψ(x)→ ψ′(x) = e−iαψ(x) ' ψ(x)− iαφ(x) . (1.11)

The Lagrangian density becomes L′Dirac = LDirac+α2LDirac. Since α is infinitesimal the Lagrangiandensity will remain invariant. Using Eq.(1.7) one gets the conserved current

Jµ(x) = ψ(x)γµψ(x) . (1.12)

This group has only one generator, or charge, that can be determined as

Q =∫

d3xJ0(x) =∫d3xψ(x)γ0ψ(x)

=∫d3xψ(x)†ψ(x) . (1.13)

2The charges are in fact group representations of the generators.

7

In the case of non-Abelian groups the calculations are essentially the same. We shall illustratethis with the simple SU(2) group. One defines the fermion field multiplet, an isospin doublet,

ψ =

(ψ1

ψ2

). (1.14)

Under an SU(2) transformation the fermionic field transforms as

ψ(x)→ ψ′(x) = e−iτ.θ/2ψ(x) ' ψ(x)− i τ.θ2ψ(x) , (1.15)

where τ = (τ1, τ2, τ3) are the usual Pauli matrices, satisfying[τi2,τj2

]= iεijk

τk2, i, j, k = 1, 2, 3, (1.16)

and θ = (θ1, θ2, θ3) ∈ R3 are the SU(2) transformation parameters. In this case the free Lagrangiandensity is equal to Eq.(1.10), but the spinors have dimension two. 3 As expected, the Lagrangianis invariant under the global SU(2) symmetry, L′Dirac = LDirac +

(τ.θ2

)2 LDirac . One can determinethe currents

Jaµ = ψi(x)γµτaijψj(x) . (1.17)

To determine the charges we use the zero component of Eq.(1.17). Integrating over the spacialdimensions and using the fact that each fermion field obeys the anti-commutation relation

{ψi(x), ψj(x′)} = δijδ4(x− x′) , (1.18)

one obtains the expected relation[Qa, Qb] = iεabcQc . (1.19)

This means again that the {Qa} are generators of the SU(2) group (more detailed calculations canbe found in [5]).

1.2 Local Gauge invariance

Until now we have been discussing global symmetries, where the parameter εa of the field transfor-mation is independent of the space-time; thus fields at different space-time positions will transformby the same amount. The question that now arises is: Is it possible to extend this global symmetryand make it local, i.e., to transform the fields different in each space-time point? This is in generalnot possible: theory that is global invariant cannot be made local invariant without adding extraingredients. In fact, we shall see that these extra ingredients corresponds to the concept of forcein the context of field theory, more precisely, to a field exchange between particles when they areinteracting. To illustrate the need of these new fields we discuss below the simple cases of U(1) andSU(2), respectively, local gauge groups.

1.2.1 Abelian U(1) symmetry

Considering the Lagrangian density Eq.(1.10) for a spin-1/2 free field ψ(x) and performing a localinfinitesimal transformation

ψ′(x) = e−iα(x)ψ(x) ' ψ(x)− iα(x)ψ(x) , (1.20)3One should note that the space of the group is not the same as the space of the Dirac spinor. In the

SU(2) group one can see the spinor as a vector with two components, however, each of these components isa spinor of four components in the Poincaré group.

8

we see that the kinetic term will transform in a rather complicated way

ψ(x)∂µψ(x)→ ψ′(x)∂µψ′(x) = ψ(x)eiα(x)∂µ

(e−iα(x)ψ(x)

)= ψ(x)∂µψ(x)− iψ(x)∂µα(x)ψ(x) . (1.21)

To solve this problem one introduces the so-called gauge-covariant derivative, Dµ, such that it trans-forms similarly to the fermion fields under the local gauge transformation:

Dµψ(x)→ [Dµψ(x)]′ = e−iα(x)Dµψ(x) , (1.22)

This can be realized if we enlarge the theory with a new vector field Aµ(x), the so called gauge field,and defines the covariant derivative as

Dµψ ≡ (∂µ − ieAµ)ψ , (1.23)

where e is a free parameter which we eventually will identify with the electric charge. Then, thetransformation law for the covariant derivative Eq.(1.22) requires the gauge field transformation

Aµ(x)→ A′µ(x) = Aµ(x)− 1e∂µα(x) . (1.24)

To make the gauge field a true dynamical field we need to add to the Lagrangian its correspondingkinetic-type term. The simplest gauge-invariant term with dimension not greater four (in order topreserve renormalizability) that we are allowed to write is

LA = −14FµνF

µν , (1.25)

with

Fµν = ∂µAν − ∂νAµ . (1.26)

To verify the invariance of the antisymmetric tensor Fµν we can just use the relation between thistensor and the covariant derivative

[Dµ, Dν ]ψ = (DµDν −DνDµ)ψ ≡ ieFµνψ (1.27)

and Eq.(1.22). Combining all the terms we arrive at the Lagrangian density

L = ψiγµ(∂µ + ieAµ)ψ −mψψ − 14FµνF

µν . (1.28)

There is no mass term for the gauge field in the Lagrangian density Eq.(1.28), this is due to thefact that a term of the type AµAµ is obviously not gauge-invariant. Summing up, we started with atheory for a free electron and imposed local gauge invariance. This lead to the appearance of a gaugefield that has the same transformation rule, Eq.(1.24), as the photon in the electromagnetism. Also,the new covariant derivative, Eq.(1.23), corresponds to the familiar replacement pµ → pµ − eAµ onehas to do in order to include the interaction of ψ with the electromagnetic field. The free parameter eis now identified with the elementary charge and, similarly to what happens in the electromagnetism,the photon Aµ, is massless. Thus we can identify Eq.(1.28) as the QED Lagrangian density. 4

4 The only coupling of the photon with the electron is given by the term Dµψ which can be constructedfrom the transformation property of the electron field. Other (higher-dimensional) gauge-invariant couplingssuch as ψσµνψFµν are ruled out by the requirement of renormalizability. This is usually referred to asuniversality.

9

1.2.2 Non-Abelian SU(2) gauge symmetry

The non-Abelian local gauge symmetry, or Yang-Mills theory, was introduced by Yang and Mills in1954. Here we shall illustrate the construction for the simplest case of isospin SU(2). The fermionfield has the same representation as in Eq.(1.14). Under an SU(2) infinitesimal transformation thefermionic field transforms as

ψ(x)→ ψ′(x) = U(θ)ψ(x) ' ψ(x)− i τ.θ(x)2

ψ(x) . (1.29)

whereU(θ) = e−iτ.θ(x)/2 . (1.30)

As expected, the Lagrangian mass term is invariant, however the kinetic term transforms as

ψ(x)∂µψ(x)→ ψ′(x)∂µψ′(x) = ψ(x)∂µψ(x) + ψ(x)U−1(θ) [∂µU(θ)]ψ(x) , (1.31)

Once again, to construct a gauge-invariant Lagrangian we should follow a similar procedure to theone used in the Abelian case. We introduce the vector gauge fields Aiµ, i = 1, 2, 3 (one for each groupgenerator) to form the gauge-covariant derivative through the minimal coupling

Dµψ =(∂µ − ig

τ.Aµ

2

)ψ , (1.32)

where g is the coupling constant in analogy to e in Eq.(1.23). Demanding that Dµψ has the sametransformation property as ψ we obtain

τ.A′µ2

= U(θ)τ.Aµ

2U−1(θ)− i

g[∂µU(θ)]U−1(θ) , (1.33)

or in the infinitesimal formAi′µ = Aiµ + εijkθ

jAkµ −1g∂µθ

i . (1.34)

The second term is the transformation for a triplet (the adjoint) representation under SU(2).Thus, the Aiµ’s carry charges, in contrast to the Abelian gauge field. To obtain the antisymmetricsecond-rank tensor of the gauge fields, we start from the generalization of Eq.(1.27),

(DµDν −DνDµ)ψ ≡ ig(τ.Fµν

2

)ψ (1.35)

to get the following expression

τ.Fµν = ∂µτ.Aν − ∂ντ.Aµ − ig

2[τ.Aµ, τ.Aν ] , (1.36)

or in the infinitesimal formF iµν = ∂µA

iν − ∂νAiµ + gεijkA

jµA

kν . (1.37)

Contrarily to the Abelian case, F iµν is not invariant; it transforms as a SU(2) triplet. However,one can show that the scalar quantity F iµνF iµν is invariant, using the definition Eq.(1.35) and

[(DµDν −DνDµ)ψ]′ = U(θ)(DµDν −DνDµ)ψ .

We can now display the gauge-invariant Lagrangian density which describes the interaction betweenthe gauge fields and the SU(2) doublets

L = ψiγµDµψ −mψψ −14F iµνF

iµν . (1.38)

10

The pure Yang-Mills term, − 14F

iµνF

iµν , contains factors that are trilinear and quadrilinear in Aiµ,corresponds to the self-couplings of non-Abelian gauge fields. Note that such self coupling terms arenot present in an Abelian gauge theory. We should note that although fundamentally different fromthe Abelian case, the properties of universality and masslessness of the gauge fields remain essentiallythe same. However we should remark that in a non-Abelian gauge theory, the coupling strength g,between the gauge fields and the other fields, cannot be scaled arbitrarily, as in the Abelian gaugetheories, because of the commutation relation Eq.(1.19).

Since we have different gauge fields, so one could ask if the coupling would be different for eachone of them. If the group is simples, as the one that we worked with, there is just one couplingconstant. However, if the group is a product of some simple ones, for example SU(2) × SU(3), wewill have one coupling constant for each simple group. This is the case of the Standard Model (SM).

11

12

Chapter 2

Spontaneous Symmetry Breaking andHiggs mechanism

Contents2.1 Discrete symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Continuous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 15

The spontaneous breakdown of a continuous symmetry implies the existence of massless spinlessparticles. This was initially studied by Nambu (1960) and later by Goldstone (1961). Such scalarparticles are usually referred to as Nambu-Goldstone bosons. We shall see some illustrative examples,but first we should understand what Spontaneous Symmetry Breaking (SSB) means in our context.

If we have one element of a symmetry group, U ∈ G, which leaves H (the Hamiltonian) invariant,

UHU† = H , (2.1)

this element also connects states that form an irreducible representation of the group

U |A〉 = |B〉 . (2.2)

Immediately follows from Eq.(2.1) and Eq.(2.2) that

EA = 〈A|H |A〉 = 〈B|H |B〉 = EB . (2.3)

Thus the symmetry of the Hamiltonian H is manifest in the degeneracy of the energy eigenstatescorresponding to the irreducible representations of the symmetry group. However, in the abovestatements the invariance of the ground state under this symmetry transformation is implicit. Sincewe can construct the states |A〉 and |B〉 from the ground state |0〉 just by applying creations operatorson them, we get

U |0〉 = |0〉 . (2.4)

When Eq.(2.4) is not satisfied we loose the energy degeneracy symmetry. Such situation is com-monly referred as a spontaneous symmetry breakdown. An equivalent statement is that certain fieldsoperators have nonvanishing vacuum expectation value (VEV) 〈0|φj |0〉 6= 0. We must emphasizethat, even though the symmetry is not manifest in the degenerate energy levels, there are still sym-metry relations coming from the fact the Hamiltonian (or Lagrangian) is still invariant under thesymmetry transformation (for more details see [5]).

13

In what follows we shall analyse some cases of SSB. First we will presente a discrete symmetry, andafter an Abelian case, the group U(1), for global and local gauge symmetries using boson fields. Inthis chapter and following ones, we use boson fields whenever we are interest in braking spontaneouslya symmetry. We do that because since we are not allowed to write a potential like µ2ψψ + λ

(ψψ)2,

since we looses renormalizability.

2.1 Discrete symmetry

In this case there will not be Nambu-Goldstone bosons since we are in the presence of a discretesymmetry. However it will be studied here as a pedagogical example. We want to demonstrate underwhich circumstance the SSB takes place. Starting from the Lagrangian density for a real scalar field

L =12∂µφ∂

µφ− 12µ2φ2 − 1

4λφ4 , (2.5)

we see that it is invariant under the transformation φ → φ′ = −φ. For simplicity, let there be onlyone space dimension. We can write the Hamiltonian density, in the classical sense, as

H =12φ2 +

12

(∇φ)2 + V (φ) , (2.6)

where the potential isV (φ) =

12µ2φ2 +

14λφ4 . (2.7)

Since the first two terms of Eq.(2.6) are positively defined and since the energy should be boundedfrom below, the parameter λ must be positive. The sign of the µ2 parameter is free, so we have thetwo possible cases: µ2 > 0 and µ2 < 0, as shown in Fig-2.1. We see that for µ2 < 0 the ground statefield is nonvanishing and has the value

⟨0∣∣φ2∣∣ 0⟩ = −µ

2

λ. (2.8)

Figure 2.1: The two possible potentials regarding the sign of µ2.

In quantum field language, the ground state is the vacuum and the classical ground state fieldgiven by Eq.(2.8) corresponds to the possible VEVs of the field operator. We shall choose

〈φ〉 ≡ 〈0|φ |0〉 = v , with v =(−µ2/λ

)1/2. (2.9)

14

This is the broken symmetry condition. To verify that there is no massless Nambo-Goldstoneboson we should considerer small oscillations around the true vacuum. Thus we define a new quantumfield with vanishing VEV

φ′ = φ− v , (2.10)

the Lagrangian density becomes

L =12

(∂µφ′)−(−µ2

)φ′2 − λvφ′3 − λ

4φ′4 . (2.11)

We can conclude that φ′ describes a particle with mass(−2µ2

)1/2 and no massless particle appears.

2.2 Continuous symmetry breaking

Let us now analyse the simple case of a complex scalar field with global U(1) gauge invariance. Themost general Lagrangian density that we can write is

L = ∂µφ∗∂µφ− µ2φ∗φ− λ (φ∗φ)2

. (2.12)

This has a very similar form as the Lagrangian density in Eq.(2.5), so we can identify the potentialV (φ) as

V (φ) = µ2φ∗φ+ λ (φ∗φ)2. (2.13)

Again, we have two possibilities for the µ2 sign. Choosing µ2 < 0, as in Eq.(2.9), we get

〈φ∗φ〉 =|v|2

2, with |v|2 = −µ

2

λ. (2.14)

To determine the mass spectrum, to do that we will write the complex field in terms of two real ones,

φ =1√2

(φ1 + iφ2) , (2.15)

and choose

〈φ1〉 = v and 〈φ2〉 = 0 . (2.16)

This is completely general since the minimization of the potential only fixes the modulus. TheLagrangian density in Eq.(2.12) can be written as

L =12[(∂µφ1)2 + (∂µφ2)2

]− µ2

2(φ2

1 + φ22)− λ

4(φ2

1 + φ22)2 . (2.17)

Shifting the fields

φ′1 = φ1 − v and φ′2 = φ2 , (2.18)

we write the Lagrangian density in the form

L =12[(∂µφ′1)2 + (∂µφ′2)2

]− µ2

2((φ′1 + v)2 + φ′22 )− λ

4((φ′1 + v)2 + φ′22 )2

=12[(∂µφ′1)2 + (∂µφ′2)2

]+ µ2φ′21 + constant + higher-order , (2.19)

where we just expanded the terms and used the mean field condition to get rid of the linear termsin φ1. This result allows us to state that after SSB the field φ′2 is massless. There is another way of

15

doing this calculation that turns to be easier to understand graphically. Indeed, let us parameterizethe scalar as

φ(x) = eiv ξ(x)

(v + η(x)√

2

), (2.20)

where ξ(x) and η(x) are real scalar fields, with 〈ξ〉 = 〈η〉 = 0, this is the so-called unitary gauge. TheLagrangian (2.12) can be written as

L =12∂µξ∂

νξ +12∂µη∂

νη − 12

(−2µ2)η2 + constant + higher-order . (2.21)

This Lagrangian describes two scalar fields ξ and η, one with mass mη =√−2µ2 and the other

massless, mξ = 0. This result can be easily interpreted as shown in the Fig-2.2.

Figure 2.2: This represents the shape of the potential when µ < 0. The real part Re(φ) is φ1 andIm(φ) is φ2. This can also be interpreted in terms of η and χ, where η is referred to the radialoscillations and χ to the angular oscillations.

The appearance of a massless particle is a characteristic of the spontaneous symmetry breakingand it is known as the Goldstone theorem. It can be stated as:

Let a theory be invariant under the action of a transformations group G, with n generators.Then, if the symmetry is spontaneously broken, in such a way that the vacuum stays in-variant under the action of G′, with m generators (G′ ⊂ G) zero spin particles will appear,with no mass and in equal number to the generators that do not leave the vacuum invariantunder G transformations, i.e., there will appear n−m Nambu-Goldstone bosons.

At this point one should ask: What is the advantage of having theories with SBB? They give usmassless scalar particles. Are they useful?

We saw in Chapter 1 that the imposition of a local symmetry implies the existence of masslessvector particles in equal number as the generators of the symmetry group. If we want to avoid thisfeature in the gauge theory and obtain massive vector bosons, the gauge symmetry must be brokensomehow. The reason why we have been focusing our attention to gauge theories and SSB is becausewhen we have a theory with local gauge symmetry and we spontaneously break it, the gauge bosonsacquires mass and the Nambo-Goldstone bosons disappear. In the literature this is commonly referredas the Higgs mechanism.

Let us impose local gauge invariance in the Lagrangian density (2.12), so that

L = (Dµφ)† (Dµφ)− µ2φ†φ− λ(φ†φ

)2 − 14FµνF

µν , (2.22)

16

where Dµ is defined as in Eq.(1.23), Fµν as in Eq.(1.26) and Aµ transforms as in Eq.(1.24). Whenµ2 < 0 we have SSB, meaning

⟨φ†φ

⟩=|v|2√

2, with |v| =

√−µ2

λ. (2.23)

Now we should try to see the mass spectrum. Parametrizing φ as in Eq.(2.20) the Lagrangiandensity takes the form

L = −14FµνF

µν +12∂µη∂

µη +12∂µξ∂

µξ + e2v2AµAµ + (2.24)

+√

2veAµ∂µξ + µ2η2 + interaction terms . (2.25)

From this equation it is easy to see that the field η has a mass mη = −2µ2. The term e2v2AµAµ

suggests that the gauge boson Aµ has a mass mA =√

2ev, but the presence of the term√

2veAµ∂µξ,which mixes the fields Aµ and ξ, makes this interpretation less clear. This inconvenient term can beavoid if one perform a gauge transformation with the parameter α(x)

α(x) =ξ(x)v

. (2.26)

It is the so-called unitary gauge, where the gauge boson masses can be read directly. This gaugechoice corresponds to the field transformations

φ(x) → φ′(x) =v + η(x)√

2, (2.27)

Aµ(x) → A′µ(x) = Aµ(x)− 1ev∂µξ , (2.28)

and the Lagrangian density takes the form

L = −14F ′µνF

′µν +12∂µη∂

µη + e2v2A′µA′µ − 1

2(−2µ2)η2 . (2.29)

In this gauge we can easily note that the field ξ disappears completely from the Lagrangian densityand the gauge boson Aµ acquires the mass mA =

√2ev. To understand better what happened to the

field ξ we should count the total number of degrees of freedom before and after the SSB. Before theSSB we had a complex scalar field φ, which contributes with two degrees of freedom, and one gaugeboson which was massless, having just transversal polarization states, contributing with two degreesof freedom. So before the SSB we had four degrees of freedom. After the SSB, we have one scalarfield, contributing with one degree of freedom, and a vectorial field with mass, which has now notonly transversal polarization but also longitudinal, that contributes with three degrees of freedom. Itis now clear that the degree of freedom corresponding to the scalar field ξ has been absorbed by thegauge boson Aµ as it becomes massive.

So we see that contrarily to what the Goldstone theorem states, in the case of a theory withSSB and local gauge invariance (LGI) there are no Nambu-Goldstone bosons and the vectorial fieldsacquire mass, through the Higgs mechanism.

This result can be generalized to groups of higher dimensions. Let us suppose that the Lagrangian,where now the scalar field Φ belong to a vectorial space of dimension NΦ, is invariant under atransformation group G of dimension NG. Assume that after SSB, a sub-group g of dimension ng

remains a symmetry group of the physical vacuum. If only global invariance is required we expect theappearance of (NG−ng) massless Goldstone bosons. However, if one required the initial Lagrangian to

17

be LGI under the group G, one must introduce NG gauge bosons. After SSB, from the NG generatorsonly ng remain massless, while the other (NG − ng) absorb the degrees of freedom corresponding tothe Goldstone bosons and become massive. We can now count the free parameters. Before theSSB there are NΦ massless scalar fields and NG massless gauge bosons, which contribute with NΦ

and 2NG degrees of freedom, respectively. After SSB there are NΦ− (NG−ng) massive scalar fields,(NG−ng) massive gauge bosons and ng massless gauge bosons, which contribute with NΦ−(NG−ng),3(NG−ng) and 2ng degrees of freedom, respectively. Thus, summing all, one have an identical numberof degrees after and before the SSB, and equal to NΦ + 2NG.

The last example is very simple and shows the essence of the Higgs mechanism. Unfortunately, itis too simple to be realistic in particle physics, since the field Aµ cannot be interpreted as the photonbecause of its non-zero mass. In the next chapter we will construct a much more realistic model,having an invariance under the group SU(2)⊗ U(1), the so-called electroweak theory.

18

Chapter 3

Standard Model construction

Contents3.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 Electroweak symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . 233.1.2 Charged and Neutral currents . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Fermion masses and mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Quark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Leptonic sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Accidental symmetries of the Standard Model . . . . . . . . . . . . . . . 343.4 Final remarks of the Standard Model . . . . . . . . . . . . . . . . . . . . 35

Quarks (u, d, s, c, b, t), together with leptons (e, µ, τ and their respective neutrinos νe, νµ e ντ )are the known elementary particles. The productions and decays of these particles are successfullydescribed by the electroweak standard model, an SU(2)L ⊗ U(1)Y local gauge theory. Our contem-porary view of the weak interactions is the result of a long evolution of the theoretical picture ofradioactive β-decay, n → p+ + e− + ν, introduced by Fermi in 1933, Fig-3.1. In that theory theLagrangian density was given by, [1],

LF = −GF√2

[p(x)γµn(x)] [e(x)γµν(x)] + h.c. , with GF ' 1.66371× 10−5 GeV−2 . (3.1)

Figure 3.1: Fermi theory for the neutron decay.[1]

In the following years others radioactive processes were discovered. The surprising discovery of theparity non-conservation (Lee and Yang 1956; Wu et al. 1957) stimulated a great deal of research andthe eventual formulation of the V-A theory (Feynman and Gell-Mann 1958; Sudarshan and Marshak1958; Sacurai 1958). This theory suggested an effective Lagrangian density, very similar to Eq.(3.1),to describe the weak interactions

Leff (x) = −GF√2J†µ(x)Jµ(x) + h.c. , (3.2)

19

with the weak current Jµ(x) being of the vector-axial-vector form (V-A). The discovery 1 that allthe neutrinos interacting in the weak processes had a negative helicity 2 lead to the expression forthe currents

Jµ(x) = νeαγµ (1− γ5) eα + hadronic current + h.c. . (3.3)

However, the Lagrangian density (3.2) still had some problems related with some experimentaldata and renormalizability. The solution to them was given by the Glashow-Weinberg-Salam model,that consisted in a SU(2) ⊗ U(1) with LGI and SSB. In this model we pass from a four-fermioninteraction theory to an intermediate vector boson theory (IVB), Fig-3.2.

Figure 3.2: IVB theory for the neutron decay.[1]

The ‘minimality’ of an SU(2) ⊗ U(1)−type group to ‘unify’ the weak and electromagnetic in-teractions, can be seen as follows [9] : knowing that the left-handed weak charged current and itsconjugate, by themselves, constitute two of the three generators of SU(2)L group, the only questionthat must be settled is whether the neutral non-chiral electromagnetic current is the third generatorof SU(2)L. Thus, if we choose for the left-handed ‘charge-raising’ current J†µ = 1

2 [νeγµ(1− γ5)e] (asan example), we can write the two charges Q+(t) and Q−(t) of the SU(2)L group

Q+(t) = i∫d3xJ†0(x) = 1

2

∫d3x ν†e(1− γ5)e ; Q−(t) = Q+†(t) . (3.4)

Using the anti-commutation relation for the fermions, {ψ†i (x, t), ψi(x′, t)} = δijδ3(x− x′), we get

[Q+(t), Q−(t)] = 2Q3(t) , with Q3(t) = 14

∫d3x

[ν†e(1− γ5)νe − e†(1− γ5)e

]. (3.5)

The result of Eq.(3.5) is not the electromagnetic current, since Qem =∫d3x e†e, but a neutral

chiral current. Since the Q+, Q− and Qem do not form a closed SU(2)L algebra, thus we mustenlarge the group to accommodate the non-chiral electromagnetic current. The minimal enlargementis the chiral, non-Abelian SU(2)L⊗U(1)Y group, where the Y stands for the weak hypercharge. Thenon-chiral charge Q is related to the chiral charges T3 = τ3/2 and Y by the relation

Q = T3 + Y. (3.6)

3.1 The Standard Model

The idea of using the gauge SU(2)L ⊗ U(1)Y chiral group is to spontaneously break it into theunbroken non-chiral U(1)EM group, giving mass to the IVB. For the SSB we need a field that has

1Several experiments were made to confirm the parity violation proposal of Lee and Yang. Garwin,Lederman and Weirich confirmed that through the decay process π+ → µ+ + νµ. [36]

2There is sometimes a miss understanding between chirality and helicity, which only are the same in thelimit of massless particles. In Appendix A the two definitions are studied.

20

non-zero VEV, the most economical choice being, a single complex scalar field that transforms as adoublet under SU(2)L.

We shall now introduce the Standard Model of Particle Physics, a model that is not the ultimatetheory, since it does not include gravity, but as an effective theory is in a very good agreement withthe experiment and is the basis of all particle physics. It is a model based on the local gauge groupSU(3)c⊗SU(2)L⊗U(1)Y . We have seen that the group SU(2)L⊗U(1)Y can explain the electroweaktheory. On the other hand, the SU(3)c is introduced to explain the dymamics of the quarks, i.e.strong force.

The quark model proposed by Gell-Mann and, independently, by Zweig in 1964 classifies allexisting hadrons surprisingly well based on the internal symmetry of SU(3) for hadrons composedof relatively light three quarks, and successfully explains the static properties of those particles. Inthe quark model baryons are composed of three quarks and the mesons composed of a quark and ananti-quark. Nowadays six different quarks (u, d, s, c, b, t) are known to exist and thus, it is said thatquarks possess 6 degrees of freedom called flavour. The flavour of a quark can be changed throughweak interactions mediated by charged weak bosons W±. We will discuss these bosons later in thischapter. Another degree of freedom for the quarks is the so called color, which is equivalent to theelectric charge but in the context of the strong force. The theory to describe this force is known asQuantum Chromodynamics (QCD), and it’s a gauge theory with a colour SU(3) symmetry. SinceQCD will not be of crucial importance in what we are going to do, and does not make part of the aimof these thesis, one will just accept the new gauge group regarding that it has an all theory behindit, for more information see [9]. In SM we have the following arrangement of particles

SU(2)L doublets: lαL(1, 2,−1/2) =(νeαeα

)L

, qαL(3, 2, 1/6) =(uαdα

)L

.

SU(2)L singlets: eαR (1,1,-1) , uαR (3,1,2/3) , dαR (3,1,-1/3) ,

where the index α stands for the family generation and the numbers in brackets refer to thequantum numbers under the group SU(3)c×SU(2)L⊗U(1)Y . We can now write the full electroweakLagrangian density as

L = LF + LH + LG + LY − V (φ), (3.7)

where LF is the part involving the gauge covariant derivatives of the fermions fields, LH is the partinvolving the gauge covariant derivative of the Higgs boson φ, LG is the kinetic energy contributionof the four electroweak gauge bosons, LY is the Yukawa coupling of the Higgs boson to the fermionsand V (φ) is the Higgs potential.

The Lagrangian density LF is expressed as

LF = i[lαLγ

µDµlαL + eαRγµDµeαR + qαLγ

µDµqαL + uαRγµDµuαR + dαRγ

µDµdαR], (3.8)

21

where the covariant derivatives are defined as

DµlL ≡[∂µ − ig

τk

2W kµ − ig′Y (lL)Bµ

]lL ,

DµeR ≡ [∂µ − ig′Y (eR)Bµ] eR ,

DµqL ≡[∂µ − igs

λa

2Gaµ − ig

τk

2W kµ − ig′Y (qL)Bµ

]qL , (3.9)

DµuR ≡[∂µ − igs

λa

2Gaµ − ig′Y (uR)Bµ

]uR ,

DµdR ≡[∂µ − igs

λa

2Gaµ − ig′Y (dR)Bµ

]dR ,

where λa/2 are the generators of SU(3), Gell-Mann matrices, with a = 1, ..., 8. The gauge bosons Gaµ,called gluons, are massless in the theory since SU(3) is an exact symmetry; and gs, g and g′ are thecoupling constants of SU(3)c, SU(2)L and U(1)Y , respectively. The contribution of the scalar fieldto the Lagrangian density LH is

LH = (Dµφ)† (Dµφ) , with Dµφ ≡(∂µ − ig

τk

2W kµ − ig′

Y (φ)2

Bµ

)φ (3.10)

and

φ(1, 2, 1/2) =

(φ+

φ0

). (3.11)

The purely gauge contribution of the three non-Abelian W kµ gauge fields and the single Abelian

Bµ gauge field to the Lagrangian density, LG is

LG = −14F aµνF

aµν − 14BµνB

µν , (3.12)

with

F aµν ≡ ∂µW aν − ∂νW a

µ + gεabcW bµW

cν and Bµν ≡ ∂µBν − ∂νBµ . (3.13)

One of the important assumptions that is made in constructing the standard electroweak gaugegroup is to assume that the Higgs field, whose SSB, is used to generate the finite masses for the threeweak bosons, is also used to generate finite Dirac masses for the quarks and charged leptons of thethree generations. The neutrino remains massless, we will explore this issue later. In the absence ofa neutrino contribution to LY , we get

−LY = Y lαβ lαLφeβR + Y uαβ qαLφuβR + Y dαβ qαLφdβR + h.c. , (3.14)

where

φ = iτ2φ∗ . (3.15)

This is defined in such a way to conserve the weak hypercharge. The final step in the construction ofthe electroweak Lagrangian density consists on the addition of the Higgs potential with the typicalSSB form (λ > 0, µ2 < 0)

V (φ) = µ2φ†φ+ λ(φ†φ)2 . (3.16)

22

3.1.1 Electroweak symmetry breaking

In the SM the SU(3)c group is responsible for an exact symmetry – color symmetry. Thus, SSB isonly done in the electroweak symmetry group. We recall that the minimum of V (φ) occurs as inEq.(2.23)

⟨φ†φ

⟩=|v|2

2, with |v|2 = −µ

2

λ. (3.17)

Since the Higgs doublet possesses the quantum numbers (1, 2, 1/2), we have that

(T3 + Y ) < φ >= 0 , (3.18)

which is the same as saying that (T3 + Y ) is a generator that leaves the vacuum unbroken, thusbecoming the electrical charge of the unbroken U(1)EM group as in Eq.(3.6). Before we spontaneouslybreak the gauge invariant Lagrangian density, we have a theory with one complex scalar doubletfield, four degrees of freedom, and four gauge bosons, with two degrees of freedom each, thus in thebeginning we have 4+2×4 = 12 degrees of freedom. After the SSB, since we break SU(2)L⊗U(1)Y →U(1)EM , only one generator leaves the vacuum invariant, so we would expect three Nambu-Goldstonebosons, since we are working with a local gauge group, by the Higgs mechanism, those bosons willbe absorbed by an equal number of gauge bosons that will acquire mass, W± and Z0. So in the endwe will have one real scalar field, one degree of freedom, three massive gauge bosons, three degrees offreedom each, and one massless gauge boson (photon), two degrees of freedom. Summing up, afterSSB we will have 1 + 3× 3 + 2 = 12 degrees of freedom.

One can define a new field φ′ = φ− 〈φ〉 and substitute it in the covariant derivatives, since it willbe this term which will generate the gauge bosons masses, and see that three degrees of freedom willbe absorbed by the massless gauge bosons. However one could ask which degree of freedom from thecomplex scalar fields becomes massive. To see that, one should notice that in the scalar sector, wehave

φ†φ = φ′†φ′ +⟨φ†⟩φ′ + φ′† 〈φ〉+

⟨φ†⟩〈φ〉 ,

(φ†φ)2 = v2φ′†φ′ + (⟨φ†⟩φ′ + φ′† 〈φ〉)2 + ... . (3.19)

Thus writing φ′ =

(φ′+

φ′0

), the only quadratic term in φ′ is −µ

2

4 (φ′0 + φ′0†)2, where we have used

v =√−µ2/λ. 3 What this means is that only this combination is massive , this is the physical particle

with mass m =√−2µ2. The others three states −i(φ′0−φ′0†)/2, (φ′+ +φ′+†)/2 and −i(φ′+−φ′+†)/2

are absorbed by the gauge bosons.Next we present the spectrum of gauge bosons. For that it is convenient to parameterize the

scalar doublet in terms of the fields denoting shifts from the vacuum state 〈φ〉,

φ =

(φ+

φ0

)= eiτ.ξ/v

(0

v+H√2

). (3.20)

Here the fields ξi are the so-called Nambu-Goldstone bosons being absorbed into the longitudinalcomponents ofW± and Z0 bosons, and H is the Higgs boson. All this fields, ξi and H, have zero VEV.We can rewrite the Lagrangian density in the unitary gauge, where the three Nambu-Goldstone bosons

3We are fixing the VEV to be that value. In the beginning v = eiθp−µ2/λ, therefore, we are choosing

one preferred direction and breaking spontaneously the symmetry.

23

ξi disappear by being absorbed by gauge bosons, and thus the physical spectra and their interactionsbecomes apparent. Applying the unitary SU(2) transformation, U(ξ) = e−iτ.ξ/v we will have newfields defined as

φ′ = U(ξ)φ =

(0

v+H√2

), (3.21)

l′αL = U(ξ)lαL , (3.22)

q′αL = U(ξ)qαL , (3.23)

W ′aµτa

2= U(ξ)W a

µ

τa

2U(ξ)−1 − i

g(∂µU(ξ))U†(ξ) . (3.24)

The right-handed fields and Bµ do not transform under SU(2).Let us now consider the part of the scalar field Lagrangian density in Eq.(3.10) that will give

mass to the gauge bosons via the Higgs mechanism. The Lagrangian density of the vector bosonsinteracting with Higgs, VBH, will be given by

LV BH =(

0 v+H√2

)(igτa

2W a′µ + i

g′

2B′µ

)(−ig τ

a

2W a′µ − i

g′

2B′µ

)(0

v+H√2

)

=(v +H)2

8(g2W ′aµ W

′aµ + g′2B′µB′µ − 2gg′B′µW

′3µ)=

(v +H)2

8(g2W ′1µ W

′1µ + g2W ′2µ W′2µ + (gW ′3µ − g′B′µ)2

). (3.25)

This Lagrangian density has terms corresponding to the interaction between the Higgs and thegauge bosons. The only term that does not contain the Higgs and is responsible for the gauge bosonmass is

Lmass =v2

8(g2W ′1µ W

′1µ + g2W ′2µ W′2µ + (gW ′3µ − g′B′µ)2

). (3.26)

Now, introducing the charged bosons W±µ = 1√2

(W ′1µ ± iW ′2µ

), Eq.(3.26) can be rewritten as

Lmass =14g2v2W+

µ W−µ +

18v2(gW ′3µ − g′B′µ

)2. (3.27)

The mass of the charged weak bosons is M2W = g2v2/4. The remaining term which is a combination

of the two neutral gauge fields W ′3µ and B′µ can be written as

v2

8(W ′3µ B

′µ

)( g2 −gg′

−gg′ g′2

)(W ′3µ

B′µ

), (3.28)

which can be diagonalized into

v2

8(ZµAµ)

(g2 + g′2 0

0 0

)(Zµ

Aµ

)=v2

8(g2 + g′2)ZµZµ + 0.AµAµ , (3.29)

by the orthogonal transformation(Zµ

Aµ

)=

(cos θW − sin θWsin θW cos θW

)(W ′3µ

B′µ

). (3.30)

The Z0 acquires mass, M2Z = (g2 + g′2)v2/4, and the Aµ stays massless, this one is identified as

the photon. The scalar boson (Higgs) acquires the mass m2H = −2µ2. The θW is called the Weinberg

angle and it gives the mixture between W ′3µ and B′µ. The diagonalization of Eq.(3.28) leads to

sin θW = g′√g2+g′2

, cos θW = g√g2+g′2

. (3.31)

24

From Eq.(3.31) and the values of the W± and Z0 bosons we get ρ = M2W /(M

2Z sin θW 2) = 1.

This result can be obtained using the symmetry of the Higgs doublet representation. In our choiceof the Higgs representation we use the doublet representation of a scalar field, this is the same assaying that we have four real parameters, i.e. four degrees of freedom. Then this representation hasa symmetry of a 4D Euclidean rotations, SO(4) ' SU(2)L ⊗ SU(2)R. After symmetry breaking aSU(2)D symmetry is left over and the ρ = 1 is satisfied, this is known as the custodial symmetry(see [4]). Thus, this result depends on the representation that we have choose for the Higgs boson.If one chooses a higher representation with a VEV vh and Th as being the group generators, one getthe general expression [35]

ρ =∑h v

2hT

2h⊥∑

h v2hT

2h3

, with T 2h⊥ =

12

(T 2h1 + T 2

h2) . (3.32)

Figure 3.3: Interactions between the gauge bosons Z0 and W± with the Higgs boson.[1]

If one now look back to Eq.(3.25) and write it as

LV BH = M2WW

+µ W

−µ(

1 +2vH +

H2

v2

)+

12M2ZZµZ

µ

(1 +

2vH +

H2

v2

), (3.33)

we see that the terms with the Higgs can be interpreted diagramaticaly as in Fig-3.3. One shouldnote that when we write the VBH Lagangian density for the gauge bosons there is no 1/2 factorin the W± term, due to the fact that we have two degenerated mass eigenvalues for this particlescontrarily to the Z0 boson that is unique.

The Higgs mechanism in association with the electroweak gauge group give a very precise predic-tion for the W± and Z0 masses, relating them with the VEV of the Higgs. The actual values for themeasured masses are

MW = 80.403± 0.029GeV , MZ = 91.1876± 0.0021GeV . (3.34)

From this we can get the electroweak mixing angle

sin2 θW = 1− M2W

M2Z

' 0.223 . (3.35)

One can estimate this result independently, from the decay µ− → e−νeνµ . If we assume that themoment transfer, q2, is much smaller than M2

W we can pass to four-fermion interaction, i.e. using

25

the amplitude for this process

M = −i(g√2

)2 [uνµγ

µ 12

(1− γ5)uµ

] gµν − qµqνM2W

M2W − q2

[ueγ

µ 12

(1− γ5)vνe

], (3.36)

the propagator can be expressed as

18

g2

M2W − q2

' g2

8M2W

=4πα

8 sin2 θWM2W

≡ GF√2, (3.37)

where we have used Eq.(3.47) and α = e2

4π . Using the measured value of α−1 = 137.035999710(96),the value of GF given in Eq.(3.1) and the MW , we get

sin2 θW = 0.215 , (3.38)

which is in a very good agreement with Eq.(3.35). The small difference can be understood in termsof higher-order quantum corrections. Another important result that we can get from this analysis isthe determination of the VEV from the Fermi coupling

v = (√

2GF )−1/2 ' 246GeV . (3.39)

3.1.2 Charged and Neutral currents

Figure 3.4: Charged current interaction vertices.[1]

Figure 3.5: Neutral current interaction vertices.[1]

In order to identify the currents with those of V-A theory, let us next examine the fermion gaugeinteraction as contained in Eq.(3.8). Let us insert the explicited form of the covarinat derivatives,Eq.(3.9), into the fermion Lagrangian density and look only to the terms of interactions between

26

fermions and the gauge bosons. We get the following Lagrangian density [35],[1]

−Lcurrent = lαLγµ

(gτa

2W ′aµ + g′Y (lL)Bµ

)lαL + g′eαRγ

µY (eR)BµeαR +

+qαLγµ(gτa

2W ′aµ + g′Y (qL)Bµ

)qαL + g′uαRγ

µY (uR)uαR + g′dαRγµY (dR)dαR

= g

(lαLγ

µ τa

2lαL + qαLγ

µ τa

2qαL

)W aµ +

+g′(lαLY (lL)lαL + qαLY (qL)qαL + eαRY (eR)eαR + uαRY (uR)uαR + dαRY (dR)dαR

)Bµ

= g(J1µW 1µ + J2µW 2

µ) + (gJ3µW 3µ + g′JY µBµ) , (3.40)

where we used the following definitions for the currents

Jaµ = ψγµτa

2ψ, (3.41)

JYµ = −ψ′γµY (ψ′)ψ′, (3.42)

with ψ = lL, qL, and ψ′ = lL, qL, eR, uR, dR. The two terms in Eq.(3.40) are the charged and neutralcurrents. To see that, let us start with the first term, the charged current

−LCC = g(J1µW 1µ + J2µW 2

µ)

=g√2

(J+µW+µ + J−µW−µ ) , (3.43)

where we have used the fact that J+µ = J1

µ + iJ2µ = νeαLγµeαL + uαLγµdαL ; J−µ is just the hermitian

conjugated. Thus we can write the final expression

−LCC =g√2

[νeαLγµeαL + uαLγµdαL]Wµ+ + h.c.

=g

2√

2[νeαγµ(1− γ5)eα + uαγµ(1− γ5)dα]Wµ+ + h.c. . (3.44)

This can be seen in terms of Feynman diagrams, Fig-3.4. For the neutral current in Eq.(3.40) wehave

−LNC = gJ3µW 3µ + g′JY µBµ

= gJ3µ(cos θWZµ + sin θWAµ) + tan θW g(Jemµ − J3µ)(cos θWAµ − sin θWZµ) , (3.45)

where we used Eq.(3.30), Eq.(3.31) and Eq.(3.6). This can be rearranged into

−LNC = eJemµ Aµ +g

cos θWJ0µZ

µ , (3.46)

with

e = g sin θW = g′ cos θW , (3.47)

J0µ = J3

µ − sin θW 2Jemµ . (3.48)

The neutral currents can be written out explicitly in terms of the fermion fields

J0µ =

∑f

[gfLfLγµfL + gfRfRγµfR

]=

12

∑f

[gfLfγµ(1− γ5)f + gfRfγµ(1 + γ5)f

], (3.49)

27

with f = eα, να, uα, dα and the weak neutral current couplings are gfL,R = T3(fL,R)−Q(f) sin2 θW , seeFig-3.5. This result allows us to say that the fermion replication implies, with i = L,R, gei = gµi = gτi ,gui = gci = gti , gdi = gsi = gbi , and all neutrinos with gνL = 1/2. Thus, in the SM, predictions for neutralcurrents are just given by the only one unknown parameter: the Weinberg angle θW , which reflectsthe relative coupling strength of the SU(2) and U(1) gauge groups factors. We came now to a crucialprediction of the standard electroweak theory, namely that the neutral weak current depends only onthe single parameter, sin2 θW .

Figure 3.6: Self interaction of the gauge bosons W±, Z0 and γ. Notice that there is no vertex withonly Z0, γ our both because they do not have any charge.[1]

Besides the interaction between the gauge bosons and the fermions, the bosons self interact. Thoseinteractions are generated by the kinetic Lagrangian density Eq.(3.12)and correspond to the Feynmandiagrams in Fig-3.6. One should notice that there is always at least one pair of charged gauge bosonsin the interactions, thus the electroweak theory does not allow neutral vertex between photons andthe Z0.

3.2 Fermion masses and mixing

We have previously described how theW± and Z0 bosons become massive after spontaneously break-ing SU(2)L ⊗ U(1)Y → U(1)em. Still, with no further considerations, the fermions of the theoryremain massless. The fermion assignment in the SM is given by a Dirac mass term, −mf ff =

−mf (fLfR + fRfL). Although it is invariant under U(1)em, the fermion mass term is not invariantunder SU(2)L ⊗ U(1)Y . Indeed, a fermion mass term is not a singlet under SU(2)L and besidesthe right and left handed components of f have different weak hypercharges. As a result, no purefermionic mass terms can be constructed consistently with gauge invariant principles. It is then clearthat fermion masses have to be somehow generated by the symmetry breaking mechamism. Indeed,fermion masses can arise from Yukawa interactions with the scalar Higgs doublet (cf. Eq.(3.14)).

To see how fermions acquire mass, we work again in the unitary gauge. Using Eq.(3.21) toEq.(3.23), the Lagrangian takes the form 4

−LY =(Yl)αβ√

2eαL(v +H)eβR +

(Yu)αβ√2

uαL(v +H)uβR +(Yd)αβ√

2dαL(v +H)dβR + h.c.

=v(Yl)αβ√

2eαLeβR +

v(Yu)αβ√2

uαLuβR +v(Yd)αβ√

2dαLdβR + h.c. +

+(Yl)αβ√

2HeαLeβR +

(Yu)αβ√2

HuαLuβR +(Yd)αβ√

2HdαLdβR + h.c.

= −Lmass − LHFF . (3.50)

4Note that we should use, by consistency, the primed left fields. We will ignore that, having always inmind that those are the transformed fields.

28

Figure 3.7: Interaction vertex between the Higgs boson and the fermions.[1]

The Yukawa Lagrangian density can be split into two parts, one relative to the fermion masses,Lmass, and another corresponding to the interaction of the Higgs field with the fermions, see Fig-3.7,LHFF . The matrices (Yu(d))αβ , and (Yl)αβ are the Yukawa coupling matrices for the up (down)quarks and charged leptons, respectively. In the SM, these are general complex matrices in thegeneration space giving rise to a proliferation of arbitrary parameters in the model. One can nowwrite the mass Lagrangian density in a more usual way

−Lmass = eαL(Ml)αβeβR + uαL(Mu)αβuβR + dαL(Md)αβdβR + h.c. , (3.51)

with the mass matrices

(Mu)αβ = v√2(Yu)αβ , (Md)αβ = v√

2(Yd)αβ , (Ml)αβ = v√

2(Yl)αβ . (3.52)

At this stage it is worth pointing out that, in the SM, no mass term for the neutrinos canbe constructed due to the absence of right-handed projection of the ν fields, leading to masslessneutrinos. 5 In general the quark and lepton mass matrices are complex and with no other specialproperty like being diagonal, hermitian or symmetric, i.e., the physical states do not have to coincidewith the interaction states. The physical fermion masses are the eigenvalues of the fermion massmatrices. In general, it can be mathematically shown that an arbitrary n×n complex matrix M canbe diagonalized by a bi-unitary transformation as

U†LMVR = D , (3.53)

where D is a diagonal matrix and both U and V are unitary matrices. Basically the point is thatany matrix M can always be written as the product of a hermitian matrix H and a unitary matrixT , M = HT , and the hermitian matrix H can then be diagonalized by some unitary matrix. Theproof proceeds as follows: MM† is hermitian and positive; it can be diagonalized by a unitarymatrix U , U†(MM†)U = D2, with D2 = diag(m2

1,m22, ...,m

2n). The matrix U is unique up to a

diagonal phase matrix, i.e., (UF )†(MM†)(UF ) = D2, with F = diag(eiφ1 , eiφ2 , ..., eiφn). Thesephases degrees of freedom allow to ensure that the eigenvalues of M are positive. We can define ahermitian matrix H by H = UDU†, and define T ≡ H−1M , which is a unitary matrix. So we canwrite U†HU = U†MT †U = D, or U†MV = D, with V ≡ T †U also a unitary matrix. This provesEq.(3.53).

So in order to diagonalize the mass matrices Mu, Md and M l one has to preform the unitarytransformations

uL → UuLu0L , dL → UdLd

0L , eL → U lLe

0L ,

uR → V uRu0R , dR → V dRd

0R , eR → V lRe

0R , (3.54)

5This is not the complete story, there are more complicated arguments involving accidental symmetries.We will talk more about them in further sections.

29

in such a way that we diagonalize through the bi-unitary transformations

Uu†L MuVuR = diag(mu,mc,mt) ≡ Du ,

Ud†L MdVdR = diag(md,ms,mb) ≡ Dd , (3.55)

U l†LMlVlR = diag(me,mµ,mτ ) ≡ Dl ,

the vectors e0L and similar denote the mass eigenvectors. From Eq.(3.55) it follows

Uu†L HuUuL = D2

u ,

Ud†L HdUdL = D2

d , (3.56)

U l†LHlUlL = D2

l ,

where Hi ≡MiMi†, i = u, d, l. In fact, these two systems are equivalents. In general, for matrices Mand M ′, with H = MM† and H ′ = M ′M ′† we have the equivalence

W †LMWR = M ′ ⇔ W †LHWL = H ′ , (3.57)

where WL, WR are unitary matrices. The direct implication is trivial, we just need to multiply onthe right byM ′ and use the initial relation to convert WR into WL. For the reciprocal implication wejust need to verify that the matrix WR = M†WL(M ′†)−1 is unitary and that it satisfies the relation.

So after the transformation, Eq.(3.54) the Lagrangian density in Eq.(3.51) is given by 6

−Lmass = meieiLeiR +muiuiLuiR +mdidiLdiR + h.c. . (3.58)

One can also write LHFF in a different way

LHFF = HeαL(Ml)αβv

eβR +HuαL(Mu)αβ

vuβR +HdαL

(Md)αβv

dβR + h.c. , (3.59)

performing a change of basis, putting the fermions in the mass basis we get

LHFF =mei

vHeiLeiR +H

mui

vuiLuiR +H

mdi

vdiLdiR + h.c. . (3.60)

This shows that the Higgs coupling does not mix fermion flavours and that they are proportionalto the mass of the fermion to which it couples. The matrices Mu and Md are complex 3× 3 matrices,or more general ng×ng with ng the number of generations, they contain a total of 36 real parameters,or 4n2

g. However, there is a lot of information in those matrices, due to the freedom that one has tomake weak-basis transformations (WBT).

When a theory has several fields with the same quantum numbers one is free to rewrite the La-grangian in terms of new fields, obtained from the original ones by means of a unitary transformationwhich mixes them. A WBT is a transformation of the fermion fields which leaves invariant the kineticenergy terms as well as the gauge interactions. The WBT depend on the gauge theory that one isconsidering because, if there are more gauge interactions then, in principle will be less freedom tomake WBT. In the SM we define the WBT as

qL = WLq′L ,

uR = WuRu′R , (3.61)

dR = W dRd′R ,

6From now on, unless no indices are present, we will forget the notation with the 0 for the mass eigenstatesand distinguish them from the flavour eigenstates using Latin letters for the mass and Greek ones for theflavour eigenstates.

30

where WL, WuR and W d

R are 3×3 unitary matrices acting in the family space. Therefore, under WBTthe mass matrices Mu and Md transform as

M ′u = W †LMuWuR ,

M ′d = W †LMdWdR . (3.62)

The transformed mass matrices M ′u and M ′d have the same physical content as the original ones. Tosee the usefulness of WBT let us start from a general basis where the mass matrix Mu and Md have18 free parameters each. We know from Eq.(3.53) and Eq.(3.54) that these matrices can be writtenas

Mu = UuLDuVu†R ,

Md = UdLDdVd†R . (3.63)

This is just another way to write the mass matrices, nothing in the Lagrangian density has changed.Now, one can do a WBT of the type WL = Uul , W

uR = V uR and W d

R = V dR . We get

M ′u = Du ,

M ′d = Uu†L UdLDd . (3.64)

One can ask: What is the meaning of this basis? Why should we work it and not in the other?The answer will be completely understood in the next section. However, one can accept for now thatthe matrix Uu†L UdL is the quark mixing matrix and has four degrees of freedom. One easily see that inthis basis we have 6(masses) + 4(mixing) = 10 free parameters, mush less than in the general basis.7 There are many other possibilities for the form of the mass matrices. Later in this thesis we willpresent some interesting cases. We will now go on with the study of the charged current Lagrangiandensities separatedly for quarks and leptons.

3.2.1 Quark sector

In 1963 Cabibbo introduced a ‘weak’ mixing angle with the purpose to explain the reduced transitionprobabilities for ∆Y = 1 semi-leptonic weak transitions as compared to ∆Y = 0 semi-leptonic weaktransitions. Within a few years, the Cabibbo angle approach to explain that suppression was re-expressed in terms of s and d quarks, i.e., the weak charged current of low energy phenomenology,JµCC , was described by the quark model transcription of the ‘Cabibbo’ current, thus:

JµCC ∼ uLγµdL → JµC.C. ∼ uLγ

µdθcL , (3.65)

withdθcL = dL cos θc + sL sin θc , (3.66)

θc is the Cabibbo angle, Eq.(3.65) implies that the product of two charged currents generates an‘effective’ weak neutral quark current contribution to the weak Lagrangian density LNC of the form

LqNC ∼ sin θc cos θcdLγµsL . (3.67)

These are now called flavour changing neutral current (FCNC) processes and take place in the sameorder, via the neutral weak quark currents, as the non-FCNC processes arising from the charged weak

7Actually this is the minimal free parameter that one can have, since it is equal to the physical ones. Basiswith less free parameters cannot be obtained by WBT and they would have physical implications.

31

quarks currents. In order to suppress a first-order neutral current-induced FCNC process, the GIMmechanism was introduced, whereby the orthogonal (Cabibbo-type) linear combination

sθcL = −dL sin θc + sL cos θc , (3.68)

replaces sL in all charged and neutral weak currents. The contribution of the neutral currentsdL

θcγµdθcL and sLθcγµsθcL to the neutral part of the weak Lagrangian density then becomes

LqNC ∼ g(dLθcγµdθcL + sL

θcγµsθcL )Z0µ ∼ g(dLγµdL + sLγ

µsL)Z0µ , (3.69)

which no longer contains a cross term between dL and sL of the form Eq.(3.67) and hence thereis a complete cancellation of FCNC processes. Moreover, in order to construct a charged weakcurrent with sθcL , a fourth quark, c, must be introduced to complete a "second generation" SU(2)weak isospin doublet (c, sθc), in analogy to the "first generation" weak isospin doublet (u, dθc), sothat the counterpart of JµC.C. ∼ uLγ

µdθcL is JµC.C. ∼ cLγµsθcL . Thus the GIM mechanism managed to

eliminate the entire neutral current contribution to a FCNC process and only allows it to take placeas a correction to the second order charged current transition.

This was done for the special case of two quark generations and inspired the generalization tothe Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix for an arbitrary number of quarkgenerations. Let us now see the appearance of this mixing matrix and its properties. Preforming thetransformation given in Eq.(3.54) for the quarks we get

LqCC =g√2uiLγ

µ(VCKM )ijdjLW+µ + h.c. , (3.70)

where the unitary matrix VCKM = Uu†L UdL 6= 1. From the last expression we see that in the SM theCKM mixing matrix is unitary and in general complex. However, not all the phases have physicalmeaning, since it is always possible to reduce the number of phases through the transformation

V → V ′ = P †uV Pd , (3.71)

where Pu,d are diagonal unitary matrices. Such a transformation is possible since it just correspondsto a redefinition of the fields phases that are mass eigenstates of the quarks: qi → (Pq)iiqi. Thisredefinition does not affect the diagonal mass matrices Du and Dd.

Let us now count the parameters of the CKM matrix. For a general case with n dimensions, acomplex matrix VCKM of dimension n×n will have 2n2 real parameters. Since VCKM is unitary, thereare 2

(n2

)+ n = n2 unitarity conditions. Besides, 2n− 1 phases can be cancelled trough redefinitions

of the quark fields. Therefore, the number of physical parameters in VCKM is: Nparam = 2n2 − n2 −(2n−1) = (n−1)2. Since an n×n orthogonal matrix is parametrized by Nangle = n(n−1)/2 rotationangles, we arrive to the conclusion that the number of physical phases is: Nphase = Nparam−Nangle =

(n − 1)(n − 2)/2. After this parameter counting we can parameterize our CKM mixing matrix. Asimple model is the two generations case, n = 2, it follows that there is one ‘Cabibbo’ angle and zerophases so that the VCKM can be written as the real GIM-type matrix:

VCKM =

(cos θc sin θc− sin θc cos θc

). (3.72)

In a more realistic model, using three generations of quarks (the SM), we find that there are three‘Cabibbo’ angles, denoted by θ12, θ13 and θ23, and one phase, denoted by δ. Different representations

32

of the CKM mixing matrix have been proposed, one common parameterization is

VCKM = R1(θ23)Γ(δ)R2(θ13)Γ(−δ)R3(θ12)

=

d s bu [ c12c13 s12c13 s13e

iδ ]c −s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13

t s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e

iδ c23c13

(3.73)

where cij ≡ cos θij , sij ≡ sin θij , R(θij) is the rotation in the plane i− j and Γ = diag(1, 1, eiφ). Thethree sin θij are three real mixing parameters while δ is the Kobayashi-Maskawa phase. The actualmeasured values for the magnitude of the 9 CKM elements are, ref. [6],

| VCKM | =

0.97383+0.00024

−0.00023 0.2272+0.0010−0.0010

(3.96+0.09

−0.09

)× 10−3

0.2271+0.0010−0.0010 0.97296+0.00024

−0.00024

(42.21+0.10

−0.80

)× 10−3(

8.14+0.32−0.64

)× 10−3

(41.61+0.12

−0.78

)× 10−3 0.999100+0.000034

−0.000004

. (3.74)

From this parametrization one sees that the CKM parameters can be expressed as

s13 =| Vub | , s12 = |Vus|√1−|Vub|2

and s23 = |Vcb|√1−|Vub|2

. (3.75)

Therefore, to construct | VCKM | we only need to determine | Vud |, | Vus | and | Vub |. Thiswas a predictable result since | VCKM | is a unitary matrix. 8 Another, very useful, example isthe Wolfenstein parametrization, where the four mixing parameters are (λ,A, ρ, η), with λ = |Vus|,playing the role of an expansion parameter and η representing the CP violating phase

VCKM '

1− 1

2λ2 λ Aλ3(ρ− iη)

−λ 1− 12λ

2 Aλ2

Aλ3 (1− ρ− iη) −Aλ2 1

+O(λ4) , (3.76)

the experimental values for the Wolfenstein parametrization are

λ = 0.2272+0.0010−0.0010 , A = 0.818+0.007

−0.017 ,

ρ = 0.221+0.064−0.028 , η = 0.340+0.017

−0.045 , (3.77)

with ρ = ρ(1− λ2/2 + ...) and η = η(1− λ2/2 + ...). One can ask: Why did we change from an exactparametrization into some approximation? The answer is that with this parametrization one havea much more intuitively idea of what is happening in the quark sector. One see that the mixing isalmost diagonal and in leading order only the mixing between the first and third generations givesrise to CP violation. Also, the vertices amplitude in processes involving quarks are proportional toVij , so this parametrization allow us to have an idea of the magnitude of different processes in termsof the parameter of expansion, λ, see Fig-3.8.

Although there are several parametrizations, however, one can define a CP violating quantity thatis independent of the parametrization. This quantity is known as Jarlskog invariant, JCKM , and isdefined through

Im(VijVklV ∗ilV∗kj) = JCKM

3∑m,n=1

εikmεjln , (i, j, k, l = 1, 2, 3) . (3.78)

8Experimental evidences of non-unitarity would indicate that this matrix is just part of a bigger one,leading to physics beyond the SM.

33

Figure 3.8: Flavour changing transitions through the charged current coupling with the W± bosons.In this figure one easily sees that the interaction vertex in charged current processes is proportionalto the element of the mixing matrix. In the second figure one has the flavour changing amplitude.[1]

In terms of the explicit parametrizations given above, we have

JCKM=c12s12c213s13c13s23 sin δ ' λ6A2η =

(3.08+0.16

−0.18

)× 10−5 . (3.79)

This is the last observable needed to fully reconstruct VCKM , since the only unknown parameter inEq.(3.79) is the phase δ. CP violation in the quark sector is extremely challenging , however it is outof the scope of this thesis (for further reading about the subject see [3]). In this section we did theparameter counting of the CKM matrix. However, we could have done that in a more intuitive wayusing the symmetries of the problem. Working just in the quark sector, one has two Yukawa matricesgiving a total of 2× 18 = 36 parameters. Now, before writing the Yukawa mass terms for the quarksour theory had the symmetry U(3)Q × U(3)u × U(3)d, which is broken after the inclusion of thoseterms to U(1)B . Thus, the broken symmetries give 3 × 9 − 1 = 26 parameters that can be removedfrom the theory. Therefore, the number of physical parameters will be NPhys = NTotal −NBroken =

36− 26 = 10, meaning 6 masses, 3 angles and one CP phase, see [32].

3.2.2 Leptonic sector

In the leptonic sector the analysis of the mixing within the SM framework is much less elaboratedthan in the quark sector. This is mainly due to the fact that, as a consequence of the masslessness ofneutrinos, the charged currents described in Eq.(3.44) are diagonal even in the mass eigenstate basis.In order to see this, let us rotate the charged lepton fields according to Eq.(3.54). We get the chargedcurrent Lagrangian density

LlCC =g√2eiL(U lL)∗iαγµναLW

−µ + h.c. . (3.80)

On the other hand, since neutrinos are massless, one is free to rotate the left-handed neutrinofields while leaving the rest of the Lagrangian density invariant. Choosing ναL → (U lL)αjνjL theabove Lagrangian density reads

LlCC =g√2eiL(U lL)∗iα

(U lL)αjγµνjLW−µ + h.c. =g√2eiLγ

µνiLW−µ + h.c. , (3.81)

showing the absence of mixing in the leptonic sector.

3.3 Accidental symmetries of the Standard Model

In the SM there are a variety of accidental symmetries which, although not imposed at the Lagrangianlevel, appear as a consequence of Lorentz and gauge invariance, renormalisability and also of the SMparticle content. In particular, lepton (L) and baryon (B) number conservation occur due to the

34

global symmetry U(1)L ⊗ U(1)B under which the fermion fields transform as ψ → eiαL,Bψ, whereαL,B are the L and B charges. Assigning Le(νe, e−) = Lµ(νµ, µ−) = Lτ (ντ , τ−) = 1 and −1 forthe corresponding anti-particles, it is straightforward to conclude that Le, Lµ and Lτ are separatelyconserved in the SM. Therefore, lepton number is the only quantum number which differentiatesneutrinos from anti-neutrinos. Moreover, processes like ei → ej γ are forbidden since they requireLei and Lej violation.

So we could ask: What happens when one goes beyond the three level approximation? The pointis that, in the SM any of the terms that one can write in a perturbative way that violates the Lor B numbers are not Lorentz invariant and/or renormalizable, therefore B and L are automaticsymmetries of the SM at a perturbative level. Nonperturbatively, both B and L are violated by theelectroweak sphalerons 9 , however B-L remain conserved. This conserved quantum number will beof crucial importance for the construction of neutrinos mass, as we shall see in the next chapter.

3.4 Final remarks of the Standard Model

Some important properties of the SM should be emphasised so one can really understand it. In theweak charged current the interactions are mediated by the W± bosons, they exhibits the followingfeatures:

• The W± bosons couple to the fermionic doublets, where the electric charges of the two fermionpartners differ in one unit. The decay channels of the W− are then:

W− → e−νe , µ−νµ , τ

−ντ , du , sc . (3.82)

Owing to the very high mass of the top quark, mt ' 172 GeV > MW ' 80.4 GeV, its on-shellproduction through W− → bt is kinematically forbidden.The same is valid for the W+ boson.

• All fermions doublets couple to the W± bosons with the same universal strength.

• Flavor and mass eigenstates are different in quarks, thus the weak charged current for quarksmixes the mass eigenstates through the quark mixing matrix VCKM .

• The W± charged bosons couple with one or two Higgs, Z0 and γ (photon) and have also a fourvertex interaction.

The neutral current also have some features that must be mention:

• All interaction vertices are flavor conserving. Both γ an Z0 couple to a fermion and its ownanti-fermion.

• The interactions depend on the fermion electric charge Qf . Fermions with the same Qf haveexactly the same universal couplings. Neutrinos do not have electromagnetic interactions (Qν =

0), but they have non-zero coupling to the Z0, because they are SU(2)L doublets (except forthe right-handed neutrinos).

9A sphaleron is a static (time independent) solution to the electroweak field equations of the SM. Theprocess which violates the B and L cannot be represented by Feynman diagrams, and are therefore callednon-perturbative.

35

• Photons have the same interaction for both fermionic chiralities, but the Z0 couplings aredifferent for left-handed and right-handed fermions. The neutrino coupling to the Z0 involvesonly left-handed chiralities.

• The Z0 boson couples with one or two Higgs, γ does not interact with the Higgs since it doesnot have mass. There are no vertex between Z0 and γ only.

The SM has 19 arbitrary parameters. Three arbitrary gauge couplings gs, g and g′. In addition,there are 13 parameters associated with the charged lepton and quark Yukawa coupling matrices,(or, which is the same, 9 charged fermion masses, 3 mixing angles and one CP violating phase). Theremaining three parameters are the Higgs VEV v and quartic coupling λ (or instead its mass, mH),and the QCD Θ parameter. 10

The SM is a successful theory. It incorporates all the properties of the strong, weak and electro-magnetic forces and gives very good predictions for the gauge bosons, productions and decay rates,cross sections and so on. However there are some important questions that seem to indicate physicsbeyond it. First, the SM has too many arbitrary parameters and seems that the interactions aresimply put together by hand. Also, the existence of only three families, the relation between them,the relation between quarks and leptons, the possibility to unify the three couplings into one and theabsence of gravity are questions that this model does not answer.

Another very important point is the recent discovery of neutrinos oscillation which can be seenas a consequence of non-massless neutrinos. However, the SM implies that the neutrinos are exactlymassless. There are several ingredients that combine to insure it: 11

• The SM does not include fields that are singlets under the gauge group, νR(1, 1, 0). This impliesthat there are no Dirac mass terms.

• There are no scalar triplets, ∆(1, 3, 1), in the SM. Therefore, Majorana mass terms of the form∆νcLνL cannot be written.

• The SM is renormalizable. This implies that no dimension five Majorana mass terms of theform φφνcLνL are possible.

• U(1)B−L is an accidental non-anomalous global symmetry of the SM. Thus, quantum correctionscannot generate Majorana φφνcLνL mass terms.

As a final remark, one should note that both neutrinos and SU(3)c × U(1)EM gauge bosons aremassless in the SM. There is, however, a fundamental difference between these two cases. Neutrinosare massless due to an accidental symmetry, i.e not imposed by the SM, in contrast gluons and thephoton are massless due to a LGI, that is initially imposed in the SM construction. In table 3.1 theSM parameters are presented. In the next chapter we will go beyond the SM and introduce massiveneutrinos into the model.

10This parameter is responsible for CP violation in QCD. Quantum oscillations of this parameter give theaxion, a candidate for dark mater, see [33]

11We will discuss them in more detail in the next chapter.

36

Quantity Valueα 7.297 352 568(24)×10−3

Gauge coupling constants sin2 θW 0.23122(15)αs 0.1176(20)mW 80.403±0.029 GeV

Gauge Boson masses mZ 91.1876±0.0023GeVmH > 114.4GeVme 0.510 998 92±0.00000004MeVmµ 105.658 369±0.000009MeVmτ 1776.99+0.29

−0.26 MeVmu 1.5 to 3.0 MeV

Fermion masses mc 1.25±0.09 GeVmt 172.3+10.2

−7.6 GeVmd 3 to 7 MeVms 95±25 MeVmb 4.70±0.07 GeVJCKM

(3.08+0.16

−0.18

)× 10−5

CKM matrix parameters | Vud | 0.97383+0.00024−0.00023

| Vus | 0.2272+0.0010−0.0010

| Vub |(3.96+0.09

−0.09

)× 10−3

Table 3.1: The Standard Model parameters. The u, d and s quark masses are estimates of so-called‘current-quark masses’, in a mass-independent subtraction scheme such as MS at scale µ ' 2GeV.The c and b quark masses are the ‘running’ masses in the MS scheme. The top quark mass is givenby direct observations of top events.[6]

37

38

Part II

Beyond the Standard Model

39

Chapter 4

Massive Neutrinos in minimalStandard Model extensions

Contents4.1 Dirac Vs. Majorana mass term . . . . . . . . . . . . . . . . . . . . . . . . 424.2 The New-Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 The seesaw mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Leptonic mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Until now, in this thesis, we have been assuming the existence of three flavour states, the samenumber as the charged leptons. However, neutrinos are somehow ‘special’ particles and the construc-tion of the SM with three generations was an assumption. Therefore, why do we have only threeflavour neutrinos, or generations? LEP (Large Electron-Positron Collider) results came in rescue ofthe three generations. Measurements based on the Z0

µ boson width decay gave a number for thedifferent neutrino species: Nν = 2.993 ± 0.011. This result include all neutral fermions with weakcoupling with Z0

µ and mass lower than mZ0µ/2 ' 45GeV.

Our understanding of neutrinos has changed dramatically in the last decades, in great part thanksto many neutrino oscillation experiments involving solar, atmospheric, accelerator and reactor (anti)-neutrinos. In june of 1998 a very important event in neutrino physics was reported by the Super-Kamiokande collaboration, the evidence for νµ ↔ νe and νµ ↔ νe oscillations. This had already beenreported by the LSND collaboration, but Super-Kamiokande experiment was the first to show, witha high statistics, the deficit of the detected neutrino flux compared to expectations and also demon-strated that this deficit depends on the neutrino pathlength and energy in the way it is expected todepend in the case of neutrino oscillations. Since in the SM neutrinos are massless these experimentalresults were a strong evidence of physics beyond the SM.

The interest of studying neutrinos is not just the fact that it can imply physics beyond the SMbut also because neutrinos play a very important rôle in various branches of subatomic physics aswell as astrophysics and cosmology. The actual experimental upper limits for the neutrinos mass are:

• mνe < 2 eV [6](PDG 2006);

• mνµ < 0.19 MeV [6](PDG 2006);

• mντ < 18.2 MeV [6](PDG 2006);

In this chapter we shall study the Dirac an Majorana aspects of neutrino masses, how they canbe introduced into the SM and we shall finish with the analysis of the leptonic mixing.

41

4.1 Dirac Vs. Majorana mass term

In the last chapter we introduced the mass term for the fermions, coming from the Dirac Lagrangiandensity, which are called Dirac mass terms. Another possibility is a Majorana mass term, that as wewill see are not possible for charged particles, since it violates the charge, but is a valid term for theneutrinos. Before comparing this two alternatives ways one should do some important remarks.

For our discussion we will need the particle-antiparticle conjugation operator C. Its action on afermion field ψ is defined through

C : ψ → ψc = CψT, C = iγ2γ0 . (4.1)

Using these properties it is easy to see that, acting on a chiral field, C flips its chirality

C : ψL → (ψL)c = (ψc)R ≡ ψcR , ψR → (ψR)c = (ψc)L ≡ ψcL , (4.2)

i.e. the antiparticle of a left handed fermion is right handed. The particle-antiparticle conjugationoperation C must not be confused with the charge conjugation operation C which, by definition, flipsall the charge-like quantum numbers of a field (electrical charge, lepton number, baryon number,etc.) but leaves all the other quantum numbers, as chirality, intact. In particular, charge conjugationwould take a left handed neutrino into a left handed antineutrino that does not exist, which is aconsequence of the C-noninvariance of weak interactions. We are now ready to discuss the Dirac andMajorana mass terms. For massive fermions the mass term in the Lagrangian density has the form

−Lmass = mψψ = m(ψL + ψR)(ψL + ψR) = mψLψR +mψRψL , (4.3)

thus the mass term couple the right handed and left handed components of a fermion field, andtherefore a massive field must have both components. Now, there are essentially two possibilities.First, the two components are completely independent, in this case we have a Dirac field. Second,the two components are not independent and we can pass from one to the other through C acting onthem, (ψL)c = (ψc)R = ψcR = ψR, or

ψ = ψL + ηψR = ψL + η(ψL)c , (4.4)

where we have include the phase factor η = eiφ with an arbitrary phase φ. From this expressionwe can easily see that ψc = η∗ψ, meaning that particles described by Majorana fields are genuinelyneutral, i.e. their particles coincide with their antiparticles.

Resuming, a massive Dirac field needs two independent 2-components Weyl fields, ψL and ψR;together with their C-conjugates, (ψL)c = ψcR and (ψR)c = ψcL, this gives four degrees of freedom.In contrast with this, a Majorana fermion has only two degrees of freedom ψL and (ψL)c = ψcR. Formassive fermions, particle-antiparticle conjugation C and charge conjugation C coincide. For Diracfermions we have

C : ψ = ψL + ψR → (ψ)c = (ψL)c + (ψR)c = (ψc)R + (ψc)L , (4.5)

C : ψ = ψL + ψR → ψ = ψL + ψR ≡ (ψc)L + (ψc)R , (4.6)

where tilde means charge conjugation. For Majorana neutrinos, both particle-antiparticle conjugationand charge conjugation leave the field unchanged because it does not have any charge. Note that thisresult is not true when these operators act on chiral fields, has we already pointed out. For n fermion

42

species (flavours), the Majorana mass term can be written as

−LMmass =12

[(ψL)cM†ψL + ψLM(ψL)c

]=

12

[ψTLCM

†ψL + ψLCMψLT]

=12[ψTLCM

†ψL + h.c.],

(4.7)where ψ = (ψ1, ..., ψn)T is a vector in the flavour space and M is a n × n matrix. The 1/2 factorin Eq.(4.7) came from the fact that a Majorana field is constituted by two Weyl dependent spinors.Contrarily to the charged fermions, or Dirac fields, where the mass matrix has no special property,the mass matrix for a Majorana field is symmetric, Mij = Mji. 1

There is a very important difference between this two types of masses. The Dirac mass terms areinvariant under U(1) transformations, i.e. they conserve the corresponding charges (electrical charge,lepton or baryon number, etc.). However, the Majorana mass term break all the charges by twounits. This in particular means that, since the electrical charge is always conserved, charged particlescannot have Majorana mass.

4.2 The New-Standard Model

At the end of the last chapter we presented some reasons for the masslessness of the neutrino in thecontext of the SM. Now, we shall study in more detail those possible ways of extending our model.One way to go in these model constructions is start seeing the SM as an effective field theory, withnon-renormalizable operators.

A possibility is the extension of the fermionic sector to include the right handed neutrinos.This fields are singlets under SU(2)L like all other right handed fields, with the quantum nunbersνR(1, 1, 0), the Lagrangian density will have the term

Lν−SM ⊃ Y lαβlαLφνβR + h.c. , (4.8)

so this term could be the source of the neutrino mass when the symmetry is broken, however this looksvery unnatural to have only this term because it would imply Y l ' 10−12, since the experimentalbounds is mν ≤ 0.1eV. Another important remark is the fact that in this construction the neutrinosare seen as Dirac fermions and a B-L symmetry is conserved, however this is true because we did notinclude a possible mass term into the Lagrangian, 1

2MRνTRC−1νR + h.c., so in order to explain this

absence we are forced to add a symmetry to the New-Standard Model (νSM), a global U(1)B−L. 2

We can also construct a model were the mass term for the right handed neutrinos is presentand no global symmetry is needed, since this mass term is not protected by the electroweak gaugesymmetry it can have any value, this will be a very important feature for the models with the seesawmechanism. One see that there are many extensions that one can do to explain the neutrino mass,next one shall introduce the seesaw mechanism as a natural way for generation of neutrinos mass.

Another way to generate neutrino masses is to add a scalar triplet under the gauge group, leadingto an extension of the Higgs sector. The extension of the sector is done by the inclusion of a Higgstriplet ∆L in such a way that the Yukawa interaction term can be written as

Lν−SM ⊃Y ∆αβ√2

(lαL)c∆LlβL + h.c. , ∆L =

(∆0 ∆+/

√2

∆+/√

2 ∆++

). (4.9)

1To show this result we just need to use C† = CT = C−1 = −C and the fact that this term is a scalar,then the relation ψTLCM†ψL =

`ψTLCM

†ψL´T must be satiefied and we must not forget that we are working

with fermionic fields, Grassmman variables that have anticommutation relations.2One could argue that this U(1)B−L global symmetry already exist in the SM. However, that was an

accidental symmetry and since we are extending our model and have the possibility of writing a mass termfor the νR one should do unless a ‘new’ symmetry prevent it.

43

When the neutral component of ∆L acquires a vacuum expectation value < ∆0 >= vL/√

2 , aMajorana mass term of the form

Lν−SM ⊃12Y ∆αβvL(ναL)cνβL + h.c. ≡ 1

2MLαβ(ναL)cνβL + h.c. , (4.10)

is generated. Since ∆0 couples to the W and Z bosons, its VEV contributes to the masses of thesegauge bosons. Instead of the SM formulas, we now obtain

M2W =

14g2(v2 + 2v2

L) ,

M2Z =

14

(g2 + g′2)(v2 + 4v2L) , (4.11)

so thatρ ≡ M2

W

M2Z cos2 θW

=1 + 2v2

L/v22

1 + 4v2L/v

22

. (4.12)

One should notice that the extension of the Higgs sector brings to the theory an additionalhierarchy, νL � v, in order to have agreement with the experiment. This hierarchy is not solved inthe context of a extended SM. 3

The other way to generate neutrino masses is to include in the SM Lagrangian five dimensionaloperators. It was shown by Weinberg that there is only one of this operators, our Lagrangian densitywould be

Lν−SM ⊃ −λαβlαLφlβLφ

2M+O

(1M2

)+ h.c. . (4.13)

After the SSB we will get the term Lν−SM ⊃ mαβ2 νανβ , with mαβ = λαβ

v2

M . This can explainthe tiny neutrino mass if M � v. In this model the B − L symmetry is violated and neutrinos areseen as majorana particles, since this is an effective theory it is not valid for energies above at mostthe M scale. This M could be naively guess as being at the Planck scale, but that does not work. 4

Data require M < 1015 GeV, a scale that could be related with the Grand Unified Theory (GUT)scale. The following points must be regarded:

• Taking the SM to be an effective field theory implies massive neutrinos.

• Neutrino masses are small since it arises from non-renormalizable terms.

• The neutrino acquires Majorana mass.

4.3 The seesaw mechanism

The most fashionable realization of this effective theory is the seesaw mecanism. In this mechanismwe consider the existence of heavy fields, fermions or scalars, with masses of the order of ΛNP(new physics). If neutrino masses effectively arise from nonrenormalisable terms, we not only gainunderstanding regarding the existence of neutrino masses, but also of their smallness. Here, the scaleof the new physics coincides with the mass of the supper-heavy right-handed neutrino field added tothe SM.

Let us consider the mass terms that we can construct in the contest of the νSM

−LDirac = ναL(MD)αsνsR + h.c ; α = e, µ, τ ; s = s1, s2, ..., snR , (4.14)3Other theories, like SU(2)L ⊗ SU(2)R ⊗U(1)B−L or SO(10), have a more natural way of explaining this

hierarchy.4This would be a very simple way of including gravitional effects into the SM. However, mν ' v2

MPl' 10−5

eV, which is to small to explain the atmospheric neutrino deficit.

44

where MD denotes the 3 × nR Dirac neutrino mass matrix. This term can be obtain from theSM-invariant Yukawa interactions

−LYDirac = (YD)αslαLφνsR + h.c ; α = e, µ, τ ; s = s1, s2, ..., snR , (4.15)

in such a way that after SSB one get MD = vYD/√

2. As we have already seen, the presence of onlythis term makes the theory unnatural. 5 Alternatively, we can have Majorana neutrinos, leading tothe mass terms

−LLMaj =12

(ναL)c(ML)αβνβL + h.c., α, β = e, µ, τ , (4.16)

−LRMaj =12

(νsR)c(MR)ss′νs′R + h.c., s, s′ = s1, s2, ..., snR , (4.17)

where LLMaj and LRMaj are the left and right handed Majorana neutrino mass terms, respectively. Aswe have already seen the mass matrices ML and MR are symmetric, the term for the left handedneutrino came from Eq.(4.9) and the right handed term is not protected by any symmetry since it isa singlet under the gauge group.

Let us now consider a model, νSM, with the two types of mass terms, after the SSB we will havea mass Lagrangian density of the form

−Lνmass =12[νTLCM

∗LνL + νLCMLν

TL

]+

12[νTRCM

∗RνR + νRCMRν

TR

]+ νLMDνR + νRM

†DνL

=12

((νL)c νR

)(M∗L M∗D

M†D M∗R

)(νL

(νR)c

)+ h.c.

=12

(nL)cM∗nL + h.c. , (4.18)

where nL =(νL (νR)c

)Tand the neutrino mass matrixM is (3+nR)×(3+nR) complex symmetric

matrix, 3 is from the neutrino flavour states and nR from the number of right handed neutrinos. Oneshould note that the above Lagrangian density is not invariant under global phase transformation.In other words, in the general case of Dirac-Majorana mass term there are no conserved quantumnumbers that allow to distinguish a particle and its antiparticle. Therefore, it is expected thatthe fields which describe the neutrinos with definite mass are Majorana fields. To see this let usdiagonalise the mass term in Eq.(4.18) through the unitary transformation

n′L = V †nL ⇒ Lmass = − 12χdDMχ = − 1

2

∑3+nRk=1 mkχkχk, mk > 0 , (4.19)

with V a complex (3 + nR) × (3 + nR) unitary matrix, dDM = diag(m1,m2, ...,mk) the neutrinomasses and χ = (χ1, χ2, ..., χ3+nR)T = V †nL + η(V †nL)c. From this one see the fields χk havedefinite masses mk and and obey the Majorana condition (χk)c = η∗χk. One also have the mixingrelations nL = V χL and (nL)c = V ∗χR. Thus, the flavour fields νiL and νsR, with i = e, µ, τ ands = s1, s2, ..., snR , are linear unitary combinations of the left handed and right handed componentsof the Majorana neutrino mass eigenstates. Now let us proceed with the diagonalisation of M,restricting ourselves to the case case 6 nR = 3 and ML,MD �MR, to do that we shall preform the

5Unnatural in the sense that we need to impose the global U(1)B−L global symmetry.6One could ask if we really need 3 right handed neutrinos for the see saw. The answer is no, you could

have more or even less. Actually is not allowed by the experimental data to have just one right handedneutrino since it would imply two massless light ones. One could have two right handed neutrinos and thatwould imply only one massless light neutrino. However, the case with three right handed neutrinos is moreappealing from the theoretical point of view, since it gives quark-lepton symmetry. All of this in the case ofML = 0.

45

rotation nL = V n0L in such a way that

V †

(ML MD

MTD MR

)V ∗ =

(dMν

0

0 dR

), V =

(K R

S T

), (4.20)

where V is a 6 × 6 unitary matrix. Denoting by mi and Mi the light and heavy Majorana neutrinomasses, respectively, we have

dMν = diag(m1,m2,m3) , dR = diag(M1,M2,M3) . (4.21)

Since MR � ML,MD we expect the elements of R and S to be small and proportional to M−1R .

From Eq.(4.20), the following relations hold

K†MLK∗ + S†MT

DK∗ +K†MDS

∗ + S†MRS∗ = dMν

, (4.22)

K†MLR∗ + S†MT

DR∗ +K†MDT

∗ + S†MRT∗ = 0 , (4.23)

R†MLR∗ + T †MT

DR∗ +R†MDT

∗ + T †MRT∗ = dR . (4.24)

The second equality leads to

S† = −K†MDM−1R +O(MLM

−2R ,MDM

−2R ) ' −K†MDM

−1R , (4.25)

where the last approximation holds to a very high degree of accuracy. This, together with Eq.(4.22),gives

K†MνK∗ = dMν , with Mν = ML −MDM

−1R MT

D . (4.26)

The matrix Mν is the symmetric seesaw effective light neutrino mass matrix which is approxi-mately diagonalised by the matrix K. In the case under construction one has K†K ' 1. One alsoobtain

T †MRT∗ ' dR , R 'MDT

∗d−1R , (4.27)

from Eq.(4.24) and the equalityMV ∗ = V diag(dM, dR), respectively. The seesaw formula Eq.(4.26)has many interesting features that one should regard. Let us first assume that ML = 0, this is thesame as saying that we only add a right handed neutrino to the SM, the seesaw formula formula isgiven by Mν = −MDM

−1R MT

D = −Y νM−1R Y νT v2. From this expression one see that the highest MR

is the smallest mass values the left handed neutrinos will have, this allow to integrate out the heavyneutrinos and stay with an effective theory for the light ones, this is called seesaw type I. In the caseof low MR this νSM is not even a good effective theory, it gives more than three light neutrinos thatare a mixture of active neutrinos and sterile neutrinos, they are Majorana type. Another possibility,still in the type I seesaw, is to have MR = 0, 7 in this case the neutrinos are Dirac type. However,we could just add to the theory a Higgs triplet, ML 6= 0, without adding any right handed neutrinosand still have a seesaw mechanism. The seesaw formula is Mν = ML = Y ∆vL, with vL � v, thisis called seesaw type II. One can also have the two contributions at the same time to what is calledseesaw type III.

One can now see the relation between weak and mass eigenstates, for that let us recall the relationnL = V χL, one get

ναL = VαkχkL =(K R

)(ν1L ν2L ν3L N1L N2L N3L

)T, (4.28)

7This is the same of imposing lepton number conservation, more correctly B-L conservation

46

where ναL denotes the neutrino weak eigenstates with flavour i = e, ν, τ . The light and heavyneutrino mass eigenstates with mi and Ms are referred to the νiL and NsL, respectively. Because ofthe existence of 3 new mass eigenstates the leptonic charged current in the mass basis will have anextra term

LCCl = − g

2√

2(eiLKijγ

µνjL + eiLRisγµNsL)W−µ + H.c. , (4.29)

it is then clear that the matrices K and R describe the charged currents interactions of chargedleptons with light neutrinos νi and heavy neutrinos Ns, respectively. In the case when we can neglectR, the exact decoupling limit, only K is relevant for the leptonic mixing and can be identify as themixing matrix UPMNS . 8

4.4 Leptonic mixing

In this new extension of the SM we may write the leptonic mass Lagrangian density for the effectivetheory at low energies as

−Llmass =12

(ναL)c(M∗ν)αβνjL + eαL(Ml)αβeβR + H.c. , (4.30)

where M∗ν and Ml are the mass matrices for the neutrinos and charged leptons, respectively. Themass matrix M∗ν will be from now on identified as Mν ≡M∗ν , this is a complex 3× 3 symmetric, Ml

is completely arbitrary and can be diagonalised as shown in Eq.(3.55). Now for the neutrino massmatrix diagonalization we consider the following unitary transformation

ναL = UναjνjL ⇒ UνTL MνUνL = Dν = diag(m1,m2,m3), (4.31)

where mi are the neutrino mass and νi the neutrino mass eigenstates. In the mass basis both massmatrices are diagonal and the charged current interactions take the form

LCCl = − g

2√

2eiL(UPMNS)ijγµνjLW−µ + H.c. , (UPMNS)ij = (U l†L U

νL)ij . (4.32)

The parameters counting proceeds as in the quark case. However, in the leptonic sector it is onlypossible to redefine the charged leptons fields, since Majorana mass terms are not invariant underrephasing. Therefore, for the case of n generations one have a n×n matrix with 2n2 parameters, sinceit is unitary one will get n2 relations and by rephasing the charged lepton field we get n conditions,this leads to NParam = 2n2 − n2 − n = n(n− 1). Since an orthogonal n× n matrix is parameterisedby Nangles = n(n− 1)/2, one get the number of physical phases NPhase = n(n− 1)/2. For n = 3 oneget three angles and three phases to parameterise the PMNS. The standard parameterisation is

UPMNS = Uδdiag(1, eiα, eiβ), (4.33)

with

Uδ =

ν1 ν2 ν3

νe [ c12c13 s12c13 s13eiδ ]νµ −s12c23 − c12s23s13e

iδ c12c23 − s12s23s13eiδ s23c13

ντ s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e

iδ c23c13

, (4.34)

8This is true in for all the calculations presented in this thesis, so for now on when we talk about leptonicmixing one is just taking into account the first part of Eq.(4.29), since we have integrate out the massiveparticles. The capital letters mean Pontecorvo-Maki-Nakagawa-Sakata and we will study this matrix in moredetail next section.

47

where α and β are demoted by Majorana phases and δ the Dirac phase. Therefore, in the leptonicsector we have 3 CP violating phases, not one as in the quark sector. The number of physicalparameters amounts to 3 angles, 3 phases, 3 neutrino masses and 3 charged lepton masses, giving atotal of 12 physical parameters. Since me, mµ and mτ are well known parameters, there are only9 unknown physical parameters. Again we could use the symmetries of the problem to determinethe physical parameters. In the leptonic sector one has two Yukawa matrices one complex, chargedleptons, and the other symmetric, neutrinos. This leads to NTotal = 18 + 12 = 30, the Yukawaterm breaks U(3)L × U(3)e, meaning NBreak = 2× 9 = 18. Therefore, the total number of physicalparameters is Nphys = 30− 18 = 12.

We finalize this analysis with a short remark on the connection between low and high energyneutrino phenomena. Departing from the minimal extension of the νSM, type I seesaw, one has

−Llmass = νLMDνR +12

(νR)cDRνjR + eLDleR + H.c. , (4.35)

at high energies. We choose for the sake of simplicity the weak-basis where the charged leptons andthe right handed neutrinos are in the mass basis. Since the mass matrix MD is a general 3 × 3

complex matrix it has 18 parameters (9 moduli and 9 phases). However, we can redefine the lefthanded neutrino fields, thus reducing the parameters to 15 (9 moduli and 6 phases). adding the 3charged lepton masses and the 3 heavy neutrino masses, one finds 21 independent physical parameters(15 real and 6 phases), 18 of this from the neutrino sector. On the other hand, at low energy one onlyhave 9 physical parameters, this makes it almost impossible to rule out the seesaw mechanism sinceit is always possible to find a pair of MD and MR that leads to acceptable results at low energies.

Before ending this chapter one should briefly study WBT in the leptonic sector, like we did in thefor the quarks. From Eq.(4.18) one see that we only have left handed fields, thus the WBT for theleptonic fields are

n′L = WLnL ,

e′L = WLeL , (4.36)

e′R = WReR ,

leading to the mass matrices transformations

M ′l = W †LMlWR ,

M ′ν = WTLMνWL . (4.37)

Again, as in the quark sector, we have more free parameters, 30, then physical ones, 12. One cango to another basis and see that. Let us write the mass matrices as

Ml = U lLDlVl†R ,

Mν = Uν∗L DνUν†L , (4.38)

Dl and Dν are real and positive. One can now preform a WBT of the type WL = U lL and W lR = U lR,

the mass matrices become

M ′l = Dl ,

M ′ν = U∗PMNSDνU†PMNS , (4.39)

which as we can see as only 12 free parameters. At this point one should regard:

48

• Right neutrino rotations are ‘frozen’, i.e. we are not free to rotate it,

• We are free to have mass eigenvalues with negative signs for the charged leptons without anyphysical meaning, since we are free to rotate the right handed fields. That is not true for theneutrinos, negative signs or other phase values will have physical meaning, and that will appearin the Majorana phases.

49

50

Chapter 5

Neutrino Oscillations

Contents5.1 Oscillation in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1.1 Two flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.1.2 Three flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2 Oscillation in matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.1 Evolution equation. Two flavour case. . . . . . . . . . . . . . . . . . . . . . 57

5.3 Experimental neutrino physics . . . . . . . . . . . . . . . . . . . . . . . . 605.3.1 Solar neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.2 Atmospheric neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Double beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Neutrino oscillations is one of the main fields of contemporary experimental and theoretical re-search in High Energy physics. The idea of neutrino oscillation was first introduced by Pontecorvo.The essence of this effect is very simple and can be found in any book of quantum mechanics. Con-sider, as an example, a two level quantum system. If the system is in one of its stationary states | ψ1 >,it will remain in that state, and the time evolution of that function is just | ψi >= e−iEit | ψi(0) >.However, if the state is not one of the Hamiltonian eigenstates, the probability to find the system inthat state will oscillate with a frequency w21 = E2 − E1, where E1 and E2 are the eigenenergies ofthe system.

In the case of neutrinos, the weak interactions create weak eigensates νe, νµ and ντ . However, as wehave seen before, in general the neutrino mass matrix is not diagonal, meaning that there are sates ν1,ν2 and ν3 that diagonalize the neutrino mass matrix and differ from the weak eigensates. Therefore,one sees that the flavour states are a linear combination of mass sates and then the probability offinding a flavour state with the same flavour, or different one, will oscillate.

Neutrino oscillation experiments have been the most important source of information on theproperties of neutrinos. Their results not only point towards the existence of massive neutrinos butalso provides a reasonable knowledge of the neutrino mass squared differences ∆m2

ij and mixingangles. 1

The available experiments concentrate their analysis on the neutrino oscillation phenomenon usingdifferent types of neutrino sources (solar, atmosferic, reactor and accelerator neutrinos). In the firsttwo cases, the comparison of the measured neutrino fluxes with the ones expected from theoreticalmodels is the main tool to probe the (θij ,∆m2

ij) parameter space. The notation adopted here will besuch that ∆m2

21 ≡ ∆m2J, ∆m232 ≡ ∆m2

atm, θ12 ≡ θJ and θ23 ≡ θamt and θ13 ≡ θCHOOZ.

1Later in this chapter we will see the importance of ∆m2ij in neutrino oscillations.

51

5.1 Oscillation in vacuum

We shall consider Dirac neutrinos oscillation in vacuum and then comment about the Majorana andDirac-Majorana cases. In the last Chapter we could have written a lagrangian density for the chargedcurrent and leptonic mass of the form

LW+mass =g

2√

2eiLγ

µ(UPMNS)ijνjLW−µ + (ml)ieiLeiR + (mD)iνiLνiR + H.c. , (5.1)

where (ml)i are the charge lepton masses and (mD)i are the Dirac neutrino masses. We can go to aWB where we have the charge leptons mass matrix and the charge current diagonal, just by redefiningthe neutrino field as νiL = (UPMNS)ijνjL. However, since the mass and flavour eigenstates for thecharged leptons are the same, this can be seen as having the charge current in the flavour basis andthe neutrino field as ναL = (UPMNS)αjνjL, from now on the UPMNS will be denoted just by U . Sowe see that the neutrino mixing matrix relates the flavour eigenstate |να〉 2 produced or absorbedalongside with the corresponding charge lepton, to the mass eigensates |νi〉, ref. [28],

|να〉 = U∗αi |νi〉 . (5.2)

Now the question that we want to answer is: Assume that at t = 0 the |να〉 state is produced, whatis the probability to find the neutrino in the state |νβ〉 at a later time t? To answer to this questionwe shall work in the mass eigensate basis. 3 In the mass eigenstate rest frame, where the propertime is τi, we have the evolution of the state given by

i∂

∂τi|νi(τi)〉 = mi |νi(τi)〉 , (5.3)

and thus|νi(τi)〉 = e−imiτi |νi(0)〉 ≡ e−imiτi |νi〉 . (5.4)

The probability amplitude of finding the neutrino at the time t in the flavour state |νβ〉 is

A(να → νβ ; t) = 〈νβ | να(τ)〉 = Uβje−imiτiU∗αi 〈νj | νi〉 = Uβje

−imjτjU∗αj . (5.5)

One can now interpret this result in a very simple way. The U∗αj = U†jα is the transformation ampli-tude of an initial flavour eigenstate να into a mass eigenstate νj , the exponential factor e−imiτi is thepropagator describing the time evolution of a mass eigenstate neutrino νj in the energy representa-tion, and the factor Uβj is the transformation amplitude of a mass eigenstate νj into the final flavoureigenstate νβ , see Fig-5.1. Thus one have the oscillation probability given by

P (να → νβ ; t) = Pνα→νβ = |A(να → νβ ; t)|2 =∣∣Uβje−imjτjU∗αj∣∣2 . (5.6)

All of this was done only for Dirac neutrinos, what about the Majorana case? In this case wewould have to change Eq.(5.1), the Dirac mass term would be replaced by a Majorana mass termmMiν

Ti Cνi + H.c. and get lepton number violation. However, we can still preform the same weak

basis transformation and obtain the same relation between flavour and mass eigenstates, Eq.(5.2).This is the same as saying that Eq.(5.6) is invariant under phase transformations, and thus doesnot depend on the CP violating Majorana phases. Therefore we conclude that there are no way todistinguish between Dirac and Majorana through neutrino oscillation.

2We lost the subscript L, this is just an abuse of notation in the neutrino oscillation framework. Everytime that we write |να〉 we mean |ναL〉.

3This basis is easier to work with since the neutrino mass eigenstates are the eigenstates of the Hamiltonian.

52

Figure 5.1: Neutrino oscillations in vacuum.[30]

The case of Dirac-Majorana neutrinos is different since, in the most general case we have a neutrinomass matrix (n+nR)× (n+nR) which gives n+nR Majorana neutrinos mass eigenstates, this meansthat our flavour state is a linear combination of light and heavy mass eigenstates. Thus in the Dirac-Majorana case besides the να → νβ oscillation we also have the possibility of να → νs, where νs isa sterile neutrino. This oscillation violates the lepton number, thus in principle one may distinguishthis case from the pure Dirac or Majorana cases. In the limit of very massive right handed neutrinosone can decouple those ones from the theory and get the same oscillation probabilities as for the othercases.

We could now start studying neutrinos oscillation for many different scenarios, however, there issomething that we should change in Eq.(5.6). This equation as the proper time of the neutrino masseigenstates, it would be better to have that expressed in terms of the time and space in the lab frame,since it is the same for all neutrinos. To do that let us assume Lorentz invariance, in that case wehave

pµxµ = (Ei, ~pi).(t, ~L) = miτi = Eit+ piL . (5.7)

Now, neutrino sources can be seen as approximately constant in time, thus for each mass eigenstate

pi =√E2 −m2

i ' E −m2i

2E, (5.8)

thus, one can write Eq.(5.7) as

miτi = Eit− piL ' Et−(E − m2

i

2E

)L

= E(t− L) +m2i

2EL. (5.9)

The first term on the right hand side, E(t−L), is an overall phase, therefore irrelevant. One can now

53

write Eq.(5.6) as

Pνα→νβ =∣∣∣∣Uβje−im2

j2E LU∗αj

∣∣∣∣2

=

∣∣∣∣∣∣δαβ +n∑j=2

UβjU∗αj

(e−i

∆m2j1

2E L − 1)∣∣∣∣∣∣

2

, (5.10)

with ∆m2ij = m2

i −m2j . 4 One can expand the modulus, for α 6= β, and get

Pνα→νβ = 4n∑j=2

|Uβj |2 |Uαj |2 sin2 ∆j1 +

+8n∑

j,k=2j>k

∣∣UβkU∗αkU∗βjUαj∣∣ sin ∆j1 sin ∆k1 cos (∆jk + δjk) . (5.11)

Where we have defined ∆ji ≡ (m2j −m2

i )L/(4E) and δjk ≡ arg(UβkU∗αkU∗βjUαj). The survival proba-

bility is given by Pνα→να = 1−∑β 6=α Pνα→νβ . This expression is very useful in the study of neutrino

and antineutrino oscillations. The reason is that, in Eq.(5.3) the only change for antineutrinos oscil-lations is the sign of the phase δjk. 5 Therefore, in the case of complex mixing matrix we will haveCP violation, Pνα→νβ 6= Pνα→νβ , see Appendix B.

5.1.1 Two flavour case

We shall now consider the neutrino oscillation in the simple case of just two neutrino species νe andνµ. The leptonic mixing matrix as the form

U =

(cos θ0 sin θ0

− sin θ0 cos θ0

), (5.12)

where θ0 the vacuum mixing angle. From Eq.(5.2) one get the relation between flavour and masseigensates. Now, using the probability formula of Eq.(5.3) one get

Pνe→νµ = sin2 2θ0 sin2

(∆m2

21

4EL

). (5.13)

The survival probability is just given by Pνe→νe = Pνµ→νµ = 1−Pνe→νµ . One can define an oscillationlength 6 as

losc =4πE

∆m221

, (5.14)

leading to an oscillation probability

Pνe→νµ = sin2 2θ0 sin2

(πL

losc

). (5.15)

This expression has two factors. The first one, sin2 2θ0, does not depend on the distance travelledby the neutrino, describing the amplitude of the neutrino oscillations. The amplitude is maximal forθ0 = π/4, maximal mixing, and minimal amplitude for θ0 = 0 or π, small mixing. The second factoroscillates with the time or distance L, 7 it allow us to conclude that, is not only necessary to have

4To derive this last expression we used the freedom of including a phase into the modulus and the orthog-onality relation

Pnk=1 UβkU

∗αk = 1.

5Note that the relation between the flavour and mass eigenstates in the case of antineutrinos is: |να〉 =Uαi |νi〉.

6The oscillation length is the distance at which the system returns to the original sate.7One should note that in this case since we are working with ultra relativistic particles t ' L.

54

large mixing, we also need an oscillation phase not to small, losc ≤ L. If losc � L the oscillationprobability undergoes fast oscillations leading to an averaged result⟨

Pνe→νµ⟩

=12

sin2 2θ0 , (5.16)

in this case only a lower bound on the mass squared difference between the two neutrinos can beobtained.

5.1.2 Three flavour case

Now we shall consider a more realistic case, where we have three flavour states like in the νSM, νe,νµ and ντ , and three mass eigenstates ν1, ν2 and ν3. The general oscillation probability is given by

Pνα→νβ = 4 |Uβ2|2 |Uα2|2 sin2 ∆21 + |Uβ3|2 |Uα3|2 sin2 ∆31

+8∣∣Uβ2U

∗α2U

∗β3Uα3

∣∣ sin ∆31 sin ∆21 cos (∆32 + δ32) . (5.17)

Now, using the standard parametrization given in Eq.(4.34) one could get the oscillation probabilitiesfor the three flavours. However, the expressions are huge, therefore it is better to study some limitcases with physical interest. Before, one should remark that from experiments the order of theneutrinos states are unknown since the experiments are only sensible to

∣∣∆m2ji

∣∣. Therefore, one hasto consider the different ways of ordering m1, m2 and m3, consistently with the data. We will use thenotation where ν1 and ν2 are the neutrinos mass states involved in solar neutrino oscillations withm1 < m2. Under this assumption one can order the neutrinos in two different ways

Normal neutrino mass spectrum (NNMS): m1 < m2 < m3 , (5.18)

Inverted neutrino mass spectrum (INMS): m3 < m1 < m2 . (5.19)

Therefore, the neutrino mass squared differences have the hierarchy

∆m221 �| ∆m2

31 |'| ∆m232 | . (5.20)

Taking as input the lightest neutrino mass m1(m3) for the NNMS (INMS), one can easily express themass of the other two neutrino states as

NNMS: m2 =√m2

1 + ∆m221, m3 =

√m2

1 + ∆m221+ | ∆m2

32 | '√m2

1+ | ∆m232 | , (5.21)

INMS: m2 =√m2

3+ | ∆m232 |, m1 =

√m2

3+ | ∆m232 | −∆m2

21 '√m2

3+ | ∆m232 | . (5.22)

Having defined the mass spectrums, see Fig-5.2, we are now ready do study some limit cases ofneutrino oscillations.

The first limiting case that one will study is of special relevance for atmospheric, reactor andaccelerator neutrino experiments. Consider the oscillations over a baselines L for which

∆m221

2EL� 1 . (5.23)

This means that the oscillations due to small mass difference ∆m221 are effectively frozen and allow

us to consider the limit ∆m221 → 0. The oscillation probability will be given by the second factor in

Eq.(5.17), thus using the standard parametrization of the mixing matrix one get

Pνe→νµ = s223 sin2 2θ13 sin2 ∆31 ,

Pνe→ντ = c223 sin2 2θ13 sin2 ∆31 , (5.24)

Pντ→νµ = c413 sin2 2θ23 sin2 ∆31 ,

55

Figure 5.2: Neutrino mass spectrums.[7]

with Pνα→νβ = Pνβ→να . The survival probability for the νe is

Pνe→νe = 1− sin2 2θ13 sin2 ∆31 , (5.25)

which coincides with the νe probability for the two flavour case. Now one shall consider anotherlimiting case, which is relevant for the solar neutrino oscillations and also long baseline reactorexperiments. In this case we consider

∆m231

2EL ' ∆m2

32

2EL� 1 . (5.26)

In this case, the oscillation due to the mass squared differences ∆m31 and ∆m32 are very fast andlead to an average effect. The survival probability is then

Pνe→νe ' c413(1− sin2 2θ12 sin2 ∆21L) + s413 . (5.27)

Finally, consider the limit Ue3 = 0 (s13 = 0), in this case one get

Pνe→νµ = c223 sin2 2θ12 sin2 ∆21 ,

Pνe→ντ = s223 sin2 2θ12 sin2 ∆21 , (5.28)

Pντ→νµ = sin2 2θ23(−s212c

212 sin2 ∆21 + s2

12 sin2 ∆31 + c212 sin2 ∆32) ,

here we did not do any assumption about the hierarchy of the mass square difference.One should note that the limit cases presented are not mutually excluding, i.e., they have some

overlap with each other. One also see that the study of two flavour case was not naive, since thatinterpretation captures almost all the physics, atmospheric neutrinos determine | ∆m2

13 | and θ13,and solar neutrinos determine | ∆m2

12 | and θ12.In general when considering the propagation of solar neutrinos in the sun or in the earth, one

should take into account matter effects. The same is true for terrestrial atmospheric neutrinos andlong baseline accelerator neutrino oscillations experiments. Therefore, next section we will introduceneutrino oscillation in matter.

56

5.2 Oscillation in matter

Neutrino oscillation in matter may differ from the oscillation in vacuum in a very significant way.When propagating in a medium, neutrinos interact with the existent particles and ‘feel’ an effectivepotential which may engender significant changes on the neutrino masses and mixing angles. Themost striking manifestation of the matter effects on neutrino oscillations is the resonance enhancementof oscillation probability, known as the Mikheyev-Smirnov-Wolfenstein (MSW) effect. In vacuumthe oscillation probability cannot exceed sin2 2θ0, therefore, small mixing angles always imply smalloscillation effect. On the contrary, matter is able to enhance neutrino mixing and therefore theneutrino oscillation probabilities can be large for small values of θ0.

What kind of effect have the matter in neutrino propagation? Neutrinos can be absorbed bythe matter constituents, or scattered off them, changing their momentum and energy. However,the probabilities of this processes, being proportional to the square of the Fermi constant, GF , aretypically very small. Neutrinos can also experience forward scattering, an elastic scattering in whichtheir momentum is not changed. This process is coherent, and it creates mean potentials Va forneutrinos which are proportional to the number densities of the scatterers. 8

5.2.1 Evolution equation. Two flavour case.

Neutrinos of all three flavours, νe, νµ and ντ , interact with the electrons, protons and neutronsof matter through neutral current (NC) interaction mediated by Z0 bosons. Electron neutrinos inaddition have charged current (CC) interactions with the electrons of the medium, which are mediatedby the W± exchange (see Fig-5.3.)

Figure 5.3: Neutrino interactions with matter.[30]

Let us consider the CC interactions. At low energies, they are described by the effective Hamil-tonian

HCC =GF√

2[eγµ(1− γ5)νe][νeγµ(1− γ5)νe] , (5.29)

where we have used the Fierz transformation. In order to obtain the coherent forward scatteringcontribution to the energy of νe in matter we fix the variables corresponding to νe and integrateover all the variables that correspond to the electron, we get Heff (νe) =< HCC >e≡ νeVeνe. Forunpolarized medium we obtain (Ve)CC ≡ VCC =

√2GFNe, where Ne is the electron number density.

Similar calculation can be done for NC contributions, giving (Va)NC = −GFNe/√

2, where Nn is the

8These potentials are of the first order in GF , but one could still expect them to be too small and of nopractical interest. However, as we will see, one has to compare the matter induced potentials of neutrinos Vawith the caracteristic neutrino kinetic energy differences ∆m2/2E. Although the potentials Va are typicallyvery small, so are ∆m2/2E, meaning that matter can strongly affect neutrino oscillations.

57

neutron number density and a = e, µ, τ . Therefore, the mater induced potential for the neutrinos is

Ve =√

2GF

(Ne −

Nn2

), Vµ = Vτ =

√2GF

(−Nn

2

). (5.30)

For antineutrinos, one has to replace Va → −Va. 9 We shall now consider the evolution of aoscillating neutrino state in matter. Contrarily to the vacuum oscillations, where we choose to workin the mass basis, for the matter oscillations it is better to work in the flavour basis since Va isdiagonal there. Let us consider the two flavour case.

In the absence of matter, the evolution equation in the mass basis is given by i(d/dt) | νi >=

Ei | νi >, thus in the flavour basis one have i(d/dt) | να >= Eαβ | νβ >= U∗αiEiUiβ | νβ >, withU∗αiEiUiβ = U∗H0U

T = H and where H0 = diag(E1, E2) is the hamiltonian in the mass basis and Hthe one in the flavour basis. Using the relativistic neutrino energy Ei ' p+m2

i /2E one obtain

id

dt

(| νe >| νµ >

)=

(p+ m12+m22

4E

)− ∆m2

4E cos 2θ0∆m2

4E sin 2θ0

∆m2

4E sin 2θ0

(p+ m12+m2

24E

)+ ∆m2

4E cos 2θ0

(| νe >| νµ >

)

=

(−∆m2

4E cos 2θ0∆m2

4E sin 2θ0

∆m2

4E sin 2θ0∆m2

4E cos 2θ0

)(| νe >| νµ >

), (5.31)

since the factor in brackets in the diagonal elements of the effective Hamiltonian coincide, theycan only modify a common phase of the neutrino states and, therefore, have no effect on neutrinooscillations. For this reason those terms were omitted in the final expression. Until now there is nopresence of matter in Eq.(5.40). To do that, one need to add the matter potential Va into Eq.(5.40),we obtain

id

dt

(| νe >| νµ >

)=

( √2GFNe ∆m2

4E sin 2θ0

∆m2

4E sin 2θ0∆m2

2E cos 2θ0

)(| νe >| νµ >

). (5.32)

This is the evolution equation which describes νe ↔ νµ oscillation in matter. Once again, wehad removed VNC from the Hamiltonian since it lead to a common phase in the flavour neutrinossates. In the two flavour approximation the evolution equation for νe ↔ ντ oscillation in matter isnot altered from the vacuum one, since Vµ = Vτ . However, in the three flavour framework matterdoes influence this oscillation because of the mixing with νe. In general the density number is notconstant, however, we will consider this simple case as an ilustration of some properties of neutrinooscillation in matter. Working in the framework where Ne = const., one know that the eigenvaluesof the hamiltonian will be the energy of each mass eigenstate in matter (νA, νB), the eigenstates ofthe matter Hamiltonian. Thus, one have an eigenvalue problem with the following solutions

E± =√

2GFNe ±∆M

2, (5.33)

with

∆M =

√(√2GFNe −

∆m2

2Ecos 2θ0

)2

+(

∆m2

2E

)2

sin2 2θ0 , (5.34)

∆M sin 2θM =∆m2

2Esin 2θ0 , (5.35)

∆M cos 2θM = A− ∆m2

2Ecos 2θ0 . (5.36)

9One should note that oscillations in mater have different probabilities for neutrinos and antineutrinos.However, that does not violate CPT , see Apendix B, since we should apply C to the matter too. Thus,neutrino oscillation probability in Earth is the same as antineutrino oscillation probbility in an Anti-Earth!

58

Thus, like in the vacuum oscillation, the neutrino flavour eigenstates in matter are a linear com-bination of mass eigenstates in matter, meaning the the two flavour case

| νe > = cos θM | ν1m > + sin θM | ν2m > ,

| νµ > = − sin θM | ν1m > + cos θM | ν2m > . (5.37)

where we made use of Eq.(5.12) where we have changed θ0 and | νi > to the matter correspondentquantity θM and | νim >, respectively. For the evolve flavour states we have

| νe(L) > = cos θMe−iE+L | ν1m > + sin θMe−iE−L | ν2m >

= cos θMe−i∆ML | ν1m > + sin θMei∆ML | ν2m > , (5.38)

| νµ(L) > = − sin θMe−iE+L | ν1m > + cos θMe−iE−L | ν2m >

= sin θMe−i∆ML | ν1m > + cos θMei∆ML | ν2m > . (5.39)

Thus the oscillation probability is given by

Pνe→νµ = sin2 2θM sin2

(∆M

2L

),

= sin2 2θM sin2

(πL

Lm

), (5.40)

just like in the vacuum case. Where we have defined

Lm =2π

∆M, (5.41)

with Lm being the matter oscillation length. The oscillation amplitude in Eq.(5.40) is given byEq.(5.36), one see that the maximal value sin2 2θM = 1 implies the condition

√2GFNe =

∆m2

2Ecos 2θ0 . (5.42)

This is called the MSW resonance condition and states that in order to satisfy Eq.(5.42) themixing angle in matter as to be maximal (θ = 45), a result independent from the mixing angle θ0.Thus, the probability of neutrino flavour oscillation in matter can be large even if the vacuum mixingangle is small. One should note that, to have a solution in Eq.(5.42) the r.h.s. must be positive,meaning

∆m2 cos 2θ0 = (m22 −m2

1)(cos2 θ0 − sin2 θ0) > 0 , (5.43)

i.e., if ν2 is heavier than neutrino ν1, one needs to have cos2 θ0 > sin2 θ0, and vise versa. In the caseof two flavours we see from Eq.(5.12) that the relation between mass and flavour eigenstate is

| ν1 > = cos θ0 | νe > − sin θ0 | νµ > ,

| ν2 > = cos θ0 | νµ > + sin θ0 | νe > . (5.44)

Thus MSW resonance condition, Eq.(5.42), requires the mass eigenstate with the lower mass to bemostly composed by νe. Thus in neutrino matter oscillations, the only operative degree of freedomis the phase. This was a very simple case, more realistic cases as adiabatic approximation and threefamilies oscillations can be seen in [8].

59

Neutrinos Experiments

Atmospheric Kamiokande[16], IMB [13], Soudan [24], K2K[14][?][?], MACRO [20] and MINOS [21]

SolarKamLAND [17], Homestake[10],Kamiokande[15], GALLEX[12], SAGE[22],Super-Kamiokande[25]and SNO[23]

Accelerators LSND[19], KARMEN [18]and MiniBooNE[29]

Reactors CHOOZ[11]

Table 5.1: Oscillation experiments

5.3 Experimental neutrino physics

The first detection of neutrinos was made in 1987, due to the 1987A supernova in the Large MagellanicCloud. Two experiments detected this phenomena, the Kamiokande, in Japan, and the IMB, in USA.The neutrinos were detected three hours earlier than the light of the explosion. This was the beginningof neutrino astrophysics.

Neutrinos have been object of intensive study, numerous experiments were, are and will be study-ing its properties, mainly oscillations. In table 5.1 we address some experiments that have been orare still working in neutrino oscillations.

5.3.1 Solar neutrinos

Figure 5.4: Energy generation in the Sun via the pp chains.[2]

Solar neutrinos are the most abundant type of neutrinos on Earth. It is believed that they area result of thermonuclear reactions inside the Sun. For stars of the same type as the Sun, the mainreactions are the fusion of hydrogen into helium

4p → α + 2 e+ + 2 νe + 28MeV . (5.45)

60

Therefore, the production of energy in the Sun is followed by the production of neutrinos. Thereaction Eq.(5.45) is a compact form of a chain reaction, see Fig-5.4 There are many different modelsfor the solar reactions, to see that in more detail look. In opposition to the photons, that take millionsof years to reach the Sun surface, the neutrinos does not feel the effects of the high matter densityand reach the surface with no difficult. This can be easily seen by a simple calculus. The typicalmatter density that neutrinos have to crossover is around 108 g/m3 and the typical νe cross sectionis around 10−47 m2. Therefore, the mean free path will be around l = (nσ)−1 ∼ 1015 m, where n isthe number of particles per volume and σ the cross section. The value of the mean free path is muchbigger than the typical radius for stars like the Sun. Therefore, neutrino scape from the Sun withoutscattering with the matter.

The predicted solar neutrino spectrum, in absence of oscillations, is shown in Fig-5.5. The reasonfor such a complex spectrum is that the overall reaction Eq.(5.45), having four particles in the initialstate, proceeds in a sequence of steps as shown in Fig-5.4.

Figure 5.5: The predicted unoscillated spectrum dΦ/dEν of solar neutrinos, together with the energythresholds of the experiments performed so far and with the best-fit oscillation survival probabilityPee(Eν) (dashed line).[2]

Many experiments have been made with the purpose of solar neutrinos study. However, as onecan see in table 5.2 that the theoretical predictions are not in agreement with the experimental data.That get known as the solar neutrino problem. The solution for this problem came only with thetheoretical model for neutrino oscillations. Therefore, if we assume that neutrinos νe can oscillate toother flavour states or even sterile ones, one is able to explain the experimental deficit.

5.3.2 Atmospheric neutrinos

The principal source of atmospheric neutrinos is the decay chain of the pions, as a consequence ofthe collisions of cosmic rays with nuclei in the atmosphere. The main reactions are

p(α, ...) + Air → π±(K±) + X

π±(K±) → µ± + νµ(νµ) (5.46)

µ± → e± + νe(νe) + νµ(νµ)

Accurate calculation of atmospheric neutrinos flux is a difficult job, the overall uncertainty is ratherlarge, and the total fluxes calculated by different authors differ by as much as 20-30%. However, the

61

Experiment Reaction ν fluxes Observed ratio Predicted ratioHomestake(1970-1994) νe

37Cl → 37Ar e mainly8B 2.56±0.23 7.6±1.3

GALLEX(1990-2003) νe

71Ga → 37Ge e all 77.5±7.7 128±9

SAGE(1991-1997) νe

71Ga → 37Ge e all 69.1±5.7 128±9

GNO(1998-2003) νe

71Ga → 37Ge e all 69.9±5.9 128±9

Kamiokande(1987-1995) ν e → ν e 8B, hep 2.80±0.36 5.05±0.9

Super-Kamiokande(1996-2001) ν e → ν e 8B, hep 2.35±0.06 5.05±0.9

SNO(1999-2003)

ν e → ν eνe d → p p eν d→ p nν

8B, hep2.31±0.211.67±0.085.17±0.38

5.05±0.95.05±0.95.05±0.9

Table 5.2: Solar neutrino experimental results. The Homestake, GALLEX, SAGE and GNO data aregiven in SNU(1SNU= 10−36 events/atom/second), the Kamiokande, Super-Kamiokande and SNOdata are measured in 106cm−2s−1.[2]

ration of the muon to electron neutrino fluxes is not to sensitive to the above uncertainties, an differentcalculations of this quantity agree to about 5%. The ratio is defined depends on the neutrino energyand the zenit angle of the neutrino trajectory, it is defined as

R(µ/e) ≡

(νµ+νµνe+νe

)data(

νµ+νµνe+νe

)MC

, (5.47)

where MC stands for Monte Carlo simulations. In table 5.3 is represented several experiments thatmeasured this ratio and in Fig-5.6 the angular distribution of the events.

Experiment Ratio R(µ/e)Kamiokande(sub-GeV) 0.60+0.07

−0.06 ± 0.05

Kamiokande(multi-GeV) 0.57+0.08

−0.07 ± 0.07

Super-Kamiokande(sub-GeV) 0.680+0.023

−0.022 ± 0.053

Super-Kamiokande(multi-GeV) 0.678+0.042

−0.039 ± 0.080

IMB 0.54± 0.05± 0.11

Soudan2 0.68± 0.11± 0.06

Table 5.3: muon-electron ratio for several experiments.[34]

These result raise a problem, why the numerical simulation differ from the experimental data?Why R(µ/e)'0.6 and not 1? And why a deficit of νµ coming from below compared with the aboveones? This get known as the atmospheric neutrino anomaly.

The solution for these these problems came once again with the inclusion of neutrino oscillationinto the theoretical model.The fact that neutrino oscillates explain why R(µ/e) 6= 1, since there will

62

Figure 5.6: The main Super-Kamiokande data: number of e± (red) and of µ± (blue) events asfunctions of direction of scattered lepton. The horizontal axis is cos θ, the cosine of the zenith angleranging between -1 (vertically up-going events) and +1 (vertically down-going events). The thirdplot shows high-energy through-going muon, only measured in the up direction. The crosses arethe data and their errors, the thin lines are the best-fit oscillation expectation, and thick lines arethe no-oscilation expectation: these are roughly up/down symmetric. Data in the multi-GeV muonsamples are very clear asymmetric, while data in the electron samples (in red) are compatible withno oscillation. [2]

be transitions from νµ to νe, ντ and also sterile. The oscillations also explain angular distribution ofthe events. If neutrinos oscillates and their propagation is done through thought matter we have totake into account the MSW effect, that will give a prediction in excellent agreement with the data.

63

5.4 Double beta decay

In some cases when the ordinary beta decay processes are energetically forbidden, the double decayprocesses, in which a nucleus A(Z,N) is converted into an isobar with the electric charge differing bytwo units, may be allowed:

A(Z,N)→ A(Z ± 2, N ∓ 2) + 2e∓ + 2νe(2νe) . (5.48)

In such decays Two neutrons of the nucleous are simultaneous converted into two protons, or viceversa. At the fundamental (quark) level, these are transitions of two d quarks into two u quarks orvice versa, see Fig. The process in Eq.(5.49) are called 2β2ν (two-neutrino double beta) decays.

Figure 5.7: (a) two-neutrino double beta decay. (b) neutrinoless double beta decay

If the lepton number is not conserved, the electron neutrino or is antineutrino emitted in on of theelementary beta decay processes forming the 2β decay can be absorbed in another, Fig-5.49, leadingto the 2β0ν (neutrinoless double beta decay):

A(Z,N)→ A(Z ± 2, N ∓ 2) + 2e∓ . (5.49)

Such processes would have a very clear experimental signature: since the recoil energy of a daugh-ter nucleus is negligibly small, the sum of the energies of the two electrons or positrons in the finalstate should be equal to the total energy release, i.e. should be represented by a discrete energy line.Therefore 2β0ν decays could serve as a sensitive probe of the lepton number violation.

2β0ν breaks not only the lepton number; since the absorbed νe or νe has the ‘wrong’ chirality, 2β0ν

also break chirality conservation. Therefore 2β0ν decay is only possible if neutrinos have nonzeroMajorana mass. Indeed, if 2β0ν decay is mediated by the νSM interactions, the amplitude of theprocess is proportional to neutrino mass:

A(2β0ν) ∝ U2eimi ≡< Mν >ee . (5.50)

One should notice that, contrarily to neutrino oscillation, in the 2β0ν the Majorana phases of theneutrinos will be important. The allowed region |(Mν)ee| as a function of the lowest mass is given inFig-5.8, this region still allow one massless neutrino.

64

Figure 5.8: Region allowed for the |(Mν)ee|.[7]

65

66

Chapter 6

Texture zeros of the fermion massmatrices

Contents6.1 Four zero schemes of quark mass matrices . . . . . . . . . . . . . . . . . 676.2 Lepton texture zero mass matrices . . . . . . . . . . . . . . . . . . . . . . 70

6.2.1 The (1,1) and off diagonal zeros . . . . . . . . . . . . . . . . . . . . . . . . . 716.2.2 Four-zero textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.3 Phenomenological implications of the four texture zero Ansätze . . . . . . . 74

The origin of mass and flavour mixing phenomena is a very challenging area of particle physics.Many physicists have devoted their time in hope to answer: What is the origin of mass? Why arethe SM masses so hierarchical? Why is quark and lepton mixing so different? In this chapter wewill focus on the masses and mixings of quarks and leptons. We have already seen, in the previouschapters, that general mass matrices have more free parameters than the physical ones. This result,points to the existence of relations among the mass matrices parameters. One possible and interestingway to reduce the number of independent parameters is to consider textures zeros on the fermionmass matrices. A more theory fundamental than the SM should be able to determine such zerosexclusively, such that the associated physical parameters could be predicted. Phenomenologically thenatural approach is to look for the simplest pattern of Mu, Md, Ml and Mν , which can result in self-consistent and experimentally-favored relations among quark and lepton masses and flavour mixingparameters. In principle one expects that zeros in the fermionic mass matrices could be related tosome discrete (or even some horizontal gauge symmetries) flavour symmetries providing useful hitstowards the dynamics of quarks and leptons mass generations and CP violation in more fundamentaltheoretical framework. However, it may happen that some of these zeros are related to WB choices,thus without any physical implications.

We will start our study by analysing texture zeros in the quark sector and extend it to the leptonsector.

6.1 Four zero schemes of quark mass matrices

In chapter 3 we saw thatMu andMd are general complex 3×3 matrices, therefore 36 free parameters.It is worth to remark that without loss of generality, the mass matrices for the quark sector can betaken to be Hermitian (the same applies to the charged lepton sector). This results from the factthat in the SM if two set of quark mass matrices (Mu,Md) and (M ′u,M

′d) are related by Eq.(3.62),

67

then they give rise to the same quark masses and charged-currents flavour mixings. Thus, one canalways use the freedom of rotating the right handed such that Mu and Md are Hermitian matrices,reducing to a total of 18 free parameters (half of the parameters that one has in general basis).

In a texture zero framework, not all of the zeros in the mass matrices have physical meaning. Toprove that let us start from a basis where Mu is diagonal and Md is Hermitian and then preform aWBT

M ′u = W †DuW,

M ′d = W †V DdV†W, (6.1)

such that (M ′u)11 = (M ′d)11 = 0. 1 As seen in ref.[37] it is always possible to find the unitary matrixW verifying these conditions, therefore the mass basis with

Mu = Md =

0 ∗ ∗∗ ∗ ∗∗ ∗ ∗

, (6.2)

has no physical meaning and the zeros in position (1,1) are called a WB-zero. A natural question thatarises at this point is whether one can get additional WB-zeros. The answer is yes, and to see thatlet us start from the last basis where the (1,1) elements of the mass matrices are zero and preform aWBT that preserve this zero and create another. Such a transformation is defined using

W =

1 0 0

0 cos θ −eiϕ sin θ

0 e−iϕ sin θ cos θ

, (6.3)

with θ and ϕ given by

tan θ =∣∣∣∣ (Md)13

(Md)12

∣∣∣∣ , ϕ = arg(Md)13 − arg(Md)12. (6.4)

Once this transformation is preformed, we are left we the following WB zeros:

M ′u =

0 ∗ ∗∗ ∗ ∗∗ ∗ ∗

, M ′d =

0 ∗ 0

∗ ∗ ∗0 ∗ ∗

. (6.5)

Exactly the same could be done for the up sector and a similar basis would be obtained where wejust need to do d↔ u in Eq.(6.4) and Eq.(6.5).

Is it possible to have the same zero structure in the up quark sector? That would lead to thefollowing structure:

M ′′u =

0 ∗ 0

∗ ∗ ∗0 ∗ ∗

, M ′′d =

0 ∗ 0

∗ ∗ ∗0 ∗ ∗

. (6.6)

This structure has the interesting features of exhibiting a parallel structure 2 and have ten freeparameters. The number of parameters can be easily determined if we note that one can make quark

1Regard that in Eq.(6.1) the left and right rotation are the same, W , in order to preserve the hermiticity.2Parallel structures are usually more common in the study of mass matrices, since we are hoping that at

high energy quarks up and down have the same origin and therefore, at low energy that is reflected in thesame mass structure.

68

redefinitions which render M ′′u real, while leaving M ′′d with two phases. However, even thought thenumber of free parameters are the same as physical ones there is no WBT that allow us to havethat structure, [37]. Therefore, the structure given in Eq.(6.6) will imply relations among physicalparameters, masses and their mixings: Thus, the four-zero texture in Eq.(6.6) is a physical basis, butis not unique. Indeed, different four-zero texture matrices with zeros located in different positionsmay have exactly the same physical content, since they can be related by a real WB transformations,therefore orthogonal, P ,

M ′u = PT Mu P ,

M ′d = PT Md P ,(6.7)

which automatically preserves the parallel structure. This is achieved by taking the six permutationsmatrices, which are isomorphic to S3 and can be realised as real matrices,

S3∼=

1 0 0

0 1 0

0 0 1

,

0 1 0

1 0 0

0 0 1

,

0 0 1

0 1 0

1 0 0

,

1 0 0

0 0 1

0 1 0

,

0 1 0

0 0 1

1 0 0

,

0 0 1

1 0 0

0 1 0

.

(6.8)

Moreover these permutations matrices lead us to a simple classification scheme. Each two-texturezero quark matrix belong to one of the following four classes:

Class I :

0 ∗ 0

∗ ∗ ∗0 ∗ ∗

0 0 ∗0 ∗ ∗∗ ∗ ∗

∗ 0 ∗0 0 ∗∗ ∗ ∗

∗ ∗ ∗∗ 0 0

∗ 0 ∗

∗ ∗ 0

∗ ∗ ∗0 ∗ 0

∗ ∗ ∗∗ ∗ 0

∗ 0 0

, (6.9)

Class II :

∗ 0 0

0 ∗ ∗0 ∗ ∗

∗ 0 ∗0 ∗ 0

∗ 0 ∗

∗ ∗ 0

∗ ∗ 0

0 0 ∗

, (6.10)

Class III :

0 ∗ ∗∗ ∗ 0

∗ 0 ∗

∗ ∗ 0

∗ 0 ∗0 ∗ ∗

∗ 0 ∗0 ∗ ∗∗ ∗ 0

, (6.11)

Class IV :

0 ∗ ∗∗ 0 ∗∗ ∗ ∗

0 ∗ ∗∗ ∗ ∗∗ ∗ 0

∗ ∗ ∗∗ 0 ∗∗ ∗ 0

. (6.12)

In principle different classes will give different physical implications, but parallel structures of acertain class will have the same physical content even their zero structure are different. Historically,this classification of parallel structures into classes came with the study of the leptonic sector. Ac-tually, the base paper, [37], which we are following in this section, does not touch in this point, it isonly study a parallel structure belonging to the class I. A more detail study of parallel structures cannow be done. Nevertheless, this is out of the scope of this thesis and should just be seen as remarkfor future work.

69

6.2 Lepton texture zero mass matrices

Since charged lepton masses are very well determined and with a strong defined hierarchy, peopleexpect that the large leptonic mixing is strongly related with the neutrino sector. Nevertheless, thestudy of parallel structures is still something that should be take into account, as we will see later inthis section. For now, we will present some features and predictions that we get by working in suchbasis.

Pattern Texture Phenomenological implications

A1

0 0 ∗0 ∗ ∗∗ ∗ ∗

m1 ' +(s12t23t13)m3eiδ

m2 ' −(s12t23/t13)m3eiδ

Rν ' s212t

223

∣∣t213 − 1/t213

∣∣A2

0 ∗ 0∗ ∗ ∗0 ∗ ∗

m1 ' −(s12t13/t23)m3eiδ

m2 ' (s12/(t13t23))m3eiδ

Rν ' s212/t

223

∣∣t213 − 1/t213

∣∣B1

∗ ∗ 0∗ 0 ∗0 ∗ ∗

m1 ' −(t223 + s12(e−iδt23 + eiδ/t23)/t13)m3

m2 ' −(t223 − s12(e−iδt23 + eiδ/t23)t13)m3

Rν ' |s12 cos δ tan 2θ23(t13 + 1/t13)|mee ' −t223

√∆m2

32/ |1− t423|

B2

∗ 0 ∗0 ∗ ∗∗ ∗ 0

m1 ' −(1/t223 + s12(e−iδ/t23 + eiδt23)t13)m3

m2 ' −(1/t223 − s12(e−iδ/t23 + eiδt23)/t13)m3

Rν ' |s12 cos δ tan 2θ23(t13 + 1/t13)|mee ' −1/t223

√∆m2

32/ |1− 1/t423|

B3

∗ 0 ∗0 0 ∗∗ ∗ ∗

m1 ' −t223(1− s12(e−iδt23 + eiδ/t23)/t13)m3

m2 ' −t223(1 + s12(e−iδt23 + eiδ/t23)t13)m3

Rν and mee are mostly the as B1

B4

∗ ∗ 0∗ ∗ ∗0 ∗ 0

m1 ' −1/t223(1− s12(e−iδ/t23 + eiδt23)t13)m3

m2 ' −1/t223(1 + s12(e−iδ/t23 + eiδt23)/t13)m3

Rν and mee are mostly the as B2

C

∗ ∗ ∗∗ 0 ∗∗ ∗ 0

|m1,2|2 ' (1 + cos2 δ tan2 2θ13)m23

|mee| =∣∣∣√∆m2

23 cot 2θ23/ cos δ∣∣∣

Table 6.1: Neutrino mass matrix patterns, which are in accord with the current experimental data.[26]

One important feature of this basis is the fact that, if the charged leptons are in the mass basis weonly have 12 free parameters that come from the neutrino mass matrix. Moreover, 12 is the numberof physical parameters in the leptonic sector, as we have seen in Eq.(4.39),

Ml =

me 0 0

0 mµ 0

0 0 mτ

, Mν = U∗

m1 0 0

0 m2 0

0 0 m3

U† . (6.13)

Therefore, any zero in the neutrino mass matrix will have a physical meaning. In the work done byPaul H. Frampton, Sheldon L. Glashow and Danny Marfatia (FGM), [26], a classification of neutrinomass matrices by classes and its physical implications are made. The patterns are accommodatedinto three classes Ai, Bi and C. For patterns of each class the phenomenological implications are

70

essential the same, only different classes have different measurable implications. It was this modelthat has motivate our study of lepton mass matrices in the context of parallel structures.

6.2.1 The (1,1) and off diagonal zeros

Having study and introduce the WBT in the quark sector, we are ready to extend it to the leptonicsector. Now, our goal is to investigate whether it is possible to find a WB transformation which,starting from arbitrary Ml, Mν leads to M ′l , M

′ν with (M ′l )11 = (M ′ν)11 = 0. Since we want to keep

M ′l hermitian, the WB transformations of Eq.(4.37) are restricted to those with WR = W :

Ml −→M ′l = W †DlW ,

Mν −→M ′ν = WT U∗Dν U†W .

(6.14)

The requirement that (M ′l )11 and (M ′ν)11 vanish leads to

me |W11|2 +mµ |W21|2 +mτ |W31|2 = 0 , (6.15)

m1X211 +m2X

221 +m3X

231 = 0 , (6.16)

where X ≡ U†W . The matrix elements X2i1 in Eq.(6.16) are then simply given as:

X2i1 = U∗ 2

1i W211 + U∗ 2

2i W221 + U∗ 2

3i W231 + 2 U∗1iW11U

∗2iW21

+ 2 U∗1iW11U∗3iW31 + 2 U∗2iW21U

∗3iW31, (i = 1, 2, 3) .

(6.17)

It is clear that in order to Eq.(6.15) to have a solution, one of the masses me, mµ, mτ have to havea opposite sign among the others. This requirement can be always fulfilled, since the sign of a Diracfermion mass can always be changed by making an appropriate chiral transformation. In order forEq.(6.16) to have a solution, the three real non-negative quantities ai ≡ |miX

2i1| should be such that

a triangle can be formed with sides a1, a2, a3. Given these non-negative numbers a1, a2, a3, it is anecessary and sufficient condition for them to be the sides of a triangle, that

2(a2

1 a22 + a2

1 a23 + a2

2 a23

)− a4

1 − a42 − a4

3 ≤ 0 . (6.18)

Given (me, mµ, mτ ), (m1, m2, m3), and U , a solution to Eqs.(6.15-6.16) maybe found through thefollowing procedure:

1. Find |W11|2, |W21|2, |W31|2 such that Eq.(6.15) is satisfied. It is clear that this is alwayspossible. One can parametrise the first column of W as

|W11|2 = c12c13 , |W11|2 = s12c13 , |W11|2 = s13 . (6.19)

Then a solution of Eq.(6.15) can be found, by adjusting θ12, θ13.

2. In order to satisfy Eq.(6.16), one has to choose W in such a way that inequality in Eq.(6.18)is satisfied. Finding a solution to Eq.(6.16) is then equivalent to the problem of finding theinternal angles of a triangle from the knowledge of its sides. If we denote ϕij ≡ arg(Xij), theinternal angles of the triangle are given by 2 (ϕ21 − ϕ11), 2 (ϕ31 − ϕ11).

Once the zero in the position (1,1) is obtained, a natural question to ask is whether one can getadditional WB zeros. It can be readily seen that there exists a second WB transformation that keeps

71

(M ′l )11 = (M ′ν)11 = 0 and leads either to (M ′l )13 = 0 or to (Mν)13 = 0. Such a transformation isdefined by

W =

1 0 0

0 cos θ −eiϕ sin θ

0 e−iϕ sin θ cos θ

, (6.20)

with θ and ϕ given by

tan θ =∣∣∣∣ (Mλ)13

(Mλ)12

∣∣∣∣ , ϕ = arg(Mλ)13 − arg(Mλ)12 , (6.21)

where λ = l or λ = ν. Similarly, it also possible to perform a WB transformation analogous to theEq.(6.20) such that one obtains the second zero in the position (1,2), in this case one has the followingrelations

tan θ =∣∣∣∣ (Mλ)12

(Mλ)13

∣∣∣∣ , ϕ = arg(Mλ)13 − arg(Mλ)12 − π , (6.22)

again, λ = l or λ = ν.

6.2.2 Four-zero textures

Until now we have seen that, similar to the quark sector, is possible to have a WB with 1 + 2 zeros.The question we address now is whether starting from arbitrary Ml, Mν leptonic matrices one canmake a WBT, so that Ml, Mν are put in the form:

Ml =

0 ∗ 0

∗ ∗ ∗0 ∗ ∗

, Mν =

0 ∗ 0

∗ ∗ ∗0 ∗ ∗

. (6.23)

Similar to the quark sector, it is not possible to go to this basis by WBT. Before giving the proof,it is useful counting the free parameters for this texture. One can make the following WBT whichkeeps the form of Ml, Mν in Eq.(6.23)

Ml → P †ϕMlPϕ , Mν → PTϕMνPϕ , (6.24)

where Pϕ ≡ diag(eiϕ1 , eiϕ2 , eiϕ3). It is trivial to verify that one can choose ϕi so that Ml becomesreal. One has still some freedom to make another WBT with Pϕ = diag(eiϕ, eiϕ, eiϕ), so that onephase in Mν is eliminated. In this way, one is left with

Ml → 4 real free parameters , (6.25)

Mν → 4 real free parameters+3 phases , (6.26)

one has altogether 4 + 4 + 3 = 11 free parameters. However, as we have already seen, the leptonicsector with 3 generations of Majorana neutrinos has 6 + 3 + 3 = 12 physical parameters only. Fromthis simple counting one sees that one looses generality by assuming the form of Eq.(6.23). 3 Letus give a ‘more’ rigorous proof. Considering U = 1, which corresponds in our notation to X = W ,then obtaining the zeros of Eq.(6.23) requires the existence of a matrix W satisfying the following

3Regard that in the quark sector the number of free parameters and physical ones are the same, thereforefor the quarks this analysis do ot lead to any conclusion.

72

four equations

meW211 +mµW

221 +mτW

231 = 0 ,

meW∗11W13 +mµW

∗21W23 +mτW

∗31W33 = 0 ,

m1W211 +m2W

221 +m3W

231 = 0 ,

m1W11W13 +m2W21W23 +m3W31W33 = 0 . (6.27)

From the 2nd and 4th equation of the system of equations (6.27), one obtains the following relation:

|W31W33| =∣∣∣∣me

mτW ∗11W13 +

mµ

mτW ∗21W23

∣∣∣∣ (6.28)

=∣∣∣∣m1

m3W11W13 +

m2

m3W21W23

∣∣∣∣ (6.29)

If in addition we consider the case me = m1 = 0, it is clear that the above equality would imply

mµ

mτ=m2

m3. (6.30)

However, as we know, a WB is only a mathematical choice with no physical content, therefore, itcannot imply any relation among physical parameters. Once more, we proved that Eq.(6.23) cannotbe obtain by WBT.

Next we consider the possibility to have four texture zero Ansätze, inspired in the FGM classifica-tion, but with equal structure in both charged lepton and neutrino sectors. When we consider theseAnzätze in the parallel context turns out to be important to organise them in various classes, showing,for example that Aq

1 and Aq2 have the same physical content while A1, A2 in [26] are different. The

systematic way of classifying such parallel structures were already done in the quark sector and nochanges are needed when we extend it to the leptonic sector. 4 One can rise the question whetheris possible that such four-zero can be compatible with type-I see-saw. In this case the general WBtransformation has the following form:

M ′l = W †MlWlR ,

M ′D = W †MDWR ,

M ′R = WTRMRWR ,

(6.31)

where W , WR and W lR are arbitrary unitary matrix. Taking the see-saw formula for the effective

Majorana left neutrino matrix, Mν ,

Mν = −M∗DM−1∗R M†D , (6.32)

it is then trivial to see that

M ′ν = −M ′ ∗D M ′ −1 ∗R M ′ †D

= −WTM∗DW∗RW

†RM

−1 ∗R W ∗RW

†RM

†DW

= WTMνW ,

(6.33)

which verifies that any WB transformation preserves the type-I see-saw formula. It can be seen fromEqs.(6.31)-(6.33) and Eq.(6.7) that if the zero structure is preserved by the see-saw then any neutrino

4One could argue that WBT in the quark and leptonic sector are not the same, they differ for the neutrinos.However, the permutation matrices have the property P † = PT , which leads to the same results.

73

matrix belonging to the same class will also preserve its zero structure and, consequently, will havethe same physical content. One can easily verify that only classes I and II have this property. Sincethe leptonic matrices belonging to Class II mix only two generations, they cannot accommodate thepresent leptonic data. Thus, in what follows we restrict our analysis to matrices in the class-I,III andIV. Although the classes III and IV do not preserve type-I see-saw formula, such matrices could arisefrom type-II see-saw.

6.2.3 Phenomenological implications of the four texture zero Ansätze

We study in this section the phenomenological implications of parallel structure Ansätze based ontexture zeros classified in the previous section. So in what follows we consider only three massmatrices respectively from I, III and IV which correspond to a representative mass matrices. Asit was mentioned before, matrices of class II are phenomenologically excluded and therefore notconsider in our analysis. We perform the analysis in two steps. The first step corresponds to the caseof factorisable leptonic mass matrices and the second step corresponds to the most general complexcase (concerning only Class I).

Factorisable leptonic mass matrices

Analytic solution are simply obtained in the case where the phases from both leptonic mass matricescan be factorised, i.e.,

Ml = K†l M0l Kl ,

Mν = KTν M

0ν Kν , (6.34)

where M0ν,l are real mass matrices and Kν,l are diagonal unitary matrices. In such a case, the PMNS

mixing matrix takes then the following general form:

U = O†`KOν , (6.35)

where the matrices O`, Oν are the real orthogonal matrices. Moreover, the phase diagonal matrixK = K†lKν can be written without loss of generality as K = diag(1, eiφ1 , eiφ2).

The full complex neutrino mass matrix of class-I

We consider in this subsection the full complex mass matrix Mν which has the form as

Mν =

0 a eiϕa 0

a eiϕa b eiϕb c eiϕc

0 c eiϕc d eiϕd

, (6.36)

with the parameters a, b, c, d and ϕa, ϕb, ϕc, ϕd taken real. It is always possible by means of a WBTto remove all the phases except one from the diagonal element (3,3). We just need to use W =

diag(ei(ϕb/2−ϕa), e−iϕb/2, ei(ϕb/2−ϕc)) and we get

M ′ν = WTMνW =

0 a 0

a b c

0 c d eiφ

, (6.37)

74

with φ = ϕb− 2ϕc +ϕd. To construct the invariants we need to define a new object, H = MM†. Wedo this to recover a hermitian object, for which we can find the invariants

Tr(Hν) = 2a2 + b2 + 2c2 + d2 = m21 +m2

2 +m23

χ(Hν) = −2c2bd cosφ+ a4 + 2a2c2 + 2a2d2 + b2d2 + c4 = m21m

22 +m2

1m23 +m2

2m23 (6.38)

det(Hν) = a4d2 = (m1m2m3)2

This equations are much more complex than the ones for the real case. Actually for Class III andIV the equations make it very difficult to obtain any numerical results. However, for the Class I thisanalysis was done. In the following discussion we will present the result of the numerical results forthe complex neutrino mss matrix in Class I, for the other classes only the factorisable case is shown.Therefore, for Class III and IV the numerical results only indicates some possible regions for theparameters analyzed, they do not give an exact prediction.

Now we proceed to the analysis of the representative mass matrices from the Classes I, III andIV. The results are summarize in Table 6.3.

Class I

M If =

0 af 0

af bf cf

0 cf df

(6.39)

In this case it is possible to determine the form of this matrix in terms of the masses and one freeparameter, df , with the help of 3 invariant quantities

Tr(Mf ) = mf1 +mf2 +mf3 ,

det(Mf ) = mf1mf2mf3 , (6.40)

χ(Mf ) = mf1mf2 +mf1mf3 +mf2mf3 ,

where mfi , i = 1, 2, 3 are the eigenvalues of the matrix (6.39). The index f designates either thecharged lepton or neutrino sectors. The element df is consider as an input parameter and can betaken positive without loss of generality. To see that let us let us suppose df < 0, then we are allowedto do a WBT in such a way that

M I′l = W †

0 al 0

al bl cl

0 cl − |dl|

W , M I′ν = WT

0 aν 0

aν bν cν

0 cν − |dν |

W . (6.41)

Defining W = diag(i, i, i), one gets

M I′l = −

0 −al 0

−al −bl −cl0 −cl |dl|

, M I′ν =

0 −aν 0

−aν −bν −cν0 −cν |dν |

. (6.42)

We can now preform another WBT, W = diag(eiπ, 1, eiπ) and get

M I′′l = −

0 al 0

al −bl cl

0 cl |dl|

, M I′′ν =

0 aν 0

aν −bν cν

0 cν |dν |

. (6.43)

75

This proves that df can be always assumed positive. One could argue about the minos sign in bf ,however, since this element is completely defined by the masses and the parameter df its sign issomehow controlled. The dependence of the matrix parameters as a function of the masses and df isgiven by

af =√−mf1mf2mf3

df,

bf = mf1 +mf2 +mf3 − df , (6.44)

cf =

√− (df −mf1)(df −mf2)(df −mf3)

df.

The necessary conditions which guarantees that the above system has solution are (with positive df ):

• mf2 < 0 implies two positive masses and mf1,3 < df < mf3,1.

• mf2 > 0 implies one negative mass. Therefore we have two possibilities:

– mf1 < 0 implying mf2,3 < df < mf3,2

– mf3 < 0 implying mf1,2 < df < mf2,1

To determine whether this Ansatz is allowed by the actual data, one needs to verify the experimen-tal boundaries expressed in table 6.2. In Fig-6.1 is represented the plots for the numerical simulationsin the case of NNMS; in a) is represented the graphic which restricts the allowed region for the lowermass, m1. One sees that m1 ≤ 10−2 eV, therefore, no lower bound is found. However, one shouldnotice that a slight improvement in the |Ue3| will immediately put a lower bound for m1. In b) ispresented the prediction for the |(Mν)ee| as a function of m1, one sees that in the allowed regionfor the lightest mass m1, |(Mν)ee| ∈

[10−3, 10−2

]eV, this region is experimental allowed as one can

see in Fig-5.8. Regarding that in the FGM Asäntze, Ai patterns predict |(Mν)ee| = 0. In contrastto the FGM case, in the context of parallel structure all these patterns have the same prediction|(Mν)ee| ∈

[10−3, 10−2

]eV.

Parameter best fit 2σ 3σ 4σ∆m2

21

[10−5eV2

]7.9 7.3-8.5 7.1-8.9 6.8-9.3

∆m232

[10−3eV2

]2.6 2.2-3.0 2.0-3.2 1.8-3.5

sin2 θ12 0.30 0.26-0.36 0.24-0.40 0.22-0.44sin2 θ23 0.50 0.38-0.63 0.34-0.68 0.31-0.71sin2 θ13 0.000 ≤ 0.025 ≤ 0.040 ≤ 0.058

Table 6.2: Best-fit values, 2σ, 3σ and 4σ intervals (1 d.o.f.) for the three-flavour neutrinos oscillationparameters from global data including solar, atmospheric, reactor (KamLAND and CHOOZ) and(K2K) experiments.[31]

In order to find some analytical expressions we must diagonalize the mass matrices. To do thatwe must have an orthogonal matrix that satisfies

OTfMfOf = diag(mf1,mf2,mf3) . (6.45)

The process to find the orthogonal matrix is rather simple. We first start note that |Mf −mfiI| = 0,which reflects the eigenvalue problem (Mf −mfiI)vi = 0. Thus, we just need to find vi in order toconstruct the orthogonal matrix. If the determinant is zero, only two rows are linearly independent

76

Figure 6.1: In this Ansatz with normal hierarchy all the experimental boundaries are verified. Infigure a) is represented the plot of |Ue3|Vs.m1 and in figure b) it is plotted the |(Mν)ee|Vs.m1, whichis a physical quantity that can be measured in the lab.

and orthogonal to vi. In a three dimensional vectorial space a way to find a vector perpendicular totwo noncolinear vectors is by using the cross product of the two vectors. In conclusion, we just needto do the cross product between two rows for each of the three matrices Mf −miI, those vectors arethe columns of the orthogonal matrix.

For this Ansatz the orthogonal matrix Of is given by

Of =

√

(df−mf1)mf2mf3(mf1−mf2)(mf1−mf3)df

√(df−mf2)mf1mf3

(mf2−mf3)(mf2−mf1)df

√(df−mf3)mf2mf1


−√− (df−mf1)mf1

(mf1−mf2)(mf1−mf3)

√− (df−mf2)mf2

(mf2−mf3)(mf2−mf1)

√− (df−mf3)mf3

(mf3−mf1)(mf3−mf2)√mf1(df−mf2)(df−mf3)

(mf1−mf2)(mf1−mf3)df−√

mf2(df−mf1)(df−mf3)(mf2−mf3)(mf2−mf1)df

√mf3(df−mf2)(df−mf1)


.

(6.46)One shall now assume dl ' mτ , which is allowed by the data, and use the strong hierarchy for thecharged leptons |me| � mµ � mτ . The orthogonal matrix is then reduced to

Ol '

1

√|me|mµ

0

−√|me|mµ

1 0

0 0 1

. (6.47)

For the neutrinos we have to be very careful with the approximations, since depending on what regionwe are working (big m1 or small m1) the validity of the approximations change. If we restrict our selfto the region m1 ∈

[10−3, 10−2

]eV, then we can get an approximated analytical expression for the

|Mee| element. To do that we start from the basis where we have a parallel structure with four zerosand diogonalises the charged leptons, the new neutrino mass matrix will be given by Mν = U†M ′νU .This basis is of great interest since the modulo of the element (Mν)ee has physical meaning, which is

(Mν)ee = m1(U∗)2e1 +m2(U∗)2

e2 +m3(U∗)2e3 . (6.48)

Therefore, the approximated analytical expression is

(Mν)ee ' 4m1m2

m23

. (6.49)

this expression give us a hint in the mass dependence of the |Mee| element.The INMS case was also studied. In Fig-6.2 b) is presented the plot of two physical quantities,

sin2 θ12 and |Ue3|, with no common region. Therefore, no physical solution for (Mν)ee was found

77

Figure 6.2: In a) is represented the |Ue3|Vs.m3, showing that only the region with high values forthe mass of m3 is allowed. In b) is plotted sin2 θ12Vs.|Ue3|, show no region allowed for this Ansatz.Therefore, these plots excludes the Ansatz of Class I with inverted hierarchy.

numerically. Analytically, we can do a similar procedure for constructing the PMNS matrix. Thenimmediately see that this is out of any experimental boundary since

m3 → 0 : |Ue3| ' 1

|Ue3| ' 0 : sin2 θ12 ' m1m1+|m2| '

12

Class III

M IIIf =

0 af cf

af bf 0

cf 0 df

(6.50)

In this class a similar analysis was done. We determined whether the Ansatz was allowed by theactual data. Again, the study was split into the cases: NNMS and INMS. Concerning the NNMS,the region found for the lowest mass was m1 ∈

[10−2, 1

]eV, as we can see in Fig-6.3 a). The region

allowed for the |(Mν)ee| was found to be within[10−2, 1

]eV. We also see a lower bound for |Ue3|.

We tried to find some analytical expression for the |(Mν)ee| as in the previous class. We followed thesame procedure, we determined the matrix entries as a function of masses and the parameter df

af =

√− (mf1 +mf2 − df )(+mf1 +mf3 − df )(mf2 +mf3 − df )

mf1 +mf2 +mf3 − 2df,

bf = mf1 +mf2 +mf3 − df ,

cf =

√(df −mf1)(df −mf2)(df −mf3)

mf1 +mf2 +mf3 − 2df.

(6.51)

Like in the previous class considered, df can always be taken positive. We construct the orthogonal

78

matrix as

(Of )11 =√

1A1

(df −mf1)(−df +mf1 +mf2)(−df +mf1 +mf3)(−df +m2 +mf3) ,

(Of )12 =√

1A2


(Of )13 =√

1A3


(Of )21 = −√− 1A1

(df −mf1)m2f1(−2df +mf1 +mf2 +mf3) ,

(Of )22 =√− 1A2

(df −mf2)m2f2(−2df +mf1 +mf2 +mf3) ,

(Of )23 =√− 1A3

(df −mf3)m2f3(−2df +mf1 +mf2 +mf3) ,

(Of )31 =√

1A1

(df −mf2)(df −mf3)(−df +mf2 +mf3)2 ,

(Of )32 = −√

1A2

(df −mf1)(df −mf3)(−df +mf1 +mf3)2 ,

(Of )33 =√

1A3

(df −mf1)(df −mf2)(−df +mf1 +mf2)2 ,

(6.52)

with

A1 = (df −mf1)(−df +mf1 +mf2)(−df +mf1 +mf3)(−df +mf2 +mf3) +

+(df −mf2)(df −mf3)(−df +mf2 +mf3)2 − (df −mf1)m2f1(−2df +mf1 +mf2 +mf3)

A2 = (df −mf1)(df −mf3)(−df +mf1 +mf3)2 + (df −mf2)(−df +mf1 +mf2)×

×(−df +mf1 +mf3)(−df +mf2 +mf3)− (df −mf2)m2f2(−2df +mf1 +mf2 +mf3)

A3 = (df −mf1)(df −mf2)(−df +mf1 +mf2)2 + (−df +mf1 +mf2)(df −mf3)×

×(−df +mf1 +mf3)(−df +mf2 +mf3)− (df −mf3)m2f3(−2df +mf1 +mf2 +mf3)

If we consider dl ' mτ and again the strong hierarchy for the charged leptons |me| � mµ � mτ , weget the same orthogonal matrix given in Eq.(6.47). In the neutrinos sector such a consideration is moreinvolved, since m1 lies in

[10−2, 1

]eV, so that the masses are almost degenerated, m1 ' m2 / m3.

Therefore, the analysis of the regions with possible solutions and the choice of a particular dν thatwould give an approximated formula for |(Mν)ee| is more difficult to deal with. Note that theorthogonal formula given in Eq.(6.52) is exact and reproduces the results stated in Fig.6.3.

A similar analysis was done for the INMS case, where we only found a upper limit for the lightmass m3, around 10−1 eV. In Fig-6.4 a) we again a lower bound for |Ue3|. In Fig-6.4 b) we present the|(Mν)ee| as a function of the lightest mass, for which it is not simple to write an analytic expressionfor |(Mν)ee|.

Class IV

mIV =

0 af bf

af 0 cf

bf cf df

(6.53)

79

Figure 6.3: In a) it is plotted |Ue3|Vs.m1, where we can see a lower bound for |Ue3|. In b) it isrepresented the |(Mν)ee|. Both plots correspond to the normal hierarchy case.

Figure 6.4: Plot of the |(Mν)ee| Vs. m3.

df = mf1 +mf2 +mf3 , (6.54)

cf = −a2f ±

√(1−m2

f1/a2f )(a2

f −m2f2)(a2

f −m2f3) + (mf2mf3 +mf1(mf2 +mf3))

2, (6.55)

bf =mf1mf2mf3 + a2

f (mf1 +mf2 +mf3)2af√cf

(6.56)

Following the same procedure as before, we studied the NNMS and INMS in the case of Class IV,presented in Fig-6.5 and Fig-6.6, respectively. Note that af acn be made always positive and istaken as an input parameter. We have a interval for the lightest mass in the normal hierarchy case,m1 ∈

[10−3, 10−1

]eV, and for the inverted hierarchy case we just have an upper limit of the order of

10−1 eV. No analytical expression for the |(Mν)ee| was found, mainly due to the choices of the mass

80

signs and values for the parameter af . The orthogonal matrix elements are now given by

(Of )11 =

√√√√√ a2f

a2f +m2

f1 +(a2f−m

2f1

c

)2 , (Of )12 =

√√√√√ a2f

a2f +m2

2 +(a2−m2

f2c

)2 ,

(Of )13 =

√√√√√ a2f

a2f +m2

f3 +(a2f−m

2f3

c

)2 , (Of )21 =

√√√√√ m2f1

a2f +m2

f1 +(a2f−m

2f1

c

)2 ,

(Of )22 =

√√√√√ m2f2

a2f +m2

f2 +(a2f−m

2f2

c

)2 , (Of )23 =

√√√√√ m2f3

a2f +m2

f3 +(a2f−m

2f3

c

)2 , (6.57)

(Of )31 =

√√√√√ (a2f −m2

f1)2

a2f +m2

f1 +(a2f−m

2f1

c

)2 , (Of )32 =

√√√√√ (a2f −m2

f2)2

a2f +m2

f2 +(a2f−m

2f2

c

)2 ,

(Of )33 =

√√√√√ (a2f −m2

f3)2

a2f +m2

f3 +(a2f−m

2f3

c

)2 .

Figure 6.5: In a) is represented the sin2 θ23Vs.m1 and in b) is represented |(Mν)ee| Vs. m1.

Figure 6.6: In a) is represented the sin2 θ12Vs.|Ue3| and in b) is represented |(Mν)ee| Vs. m3.

81

Class Normal Hierachy Inverted Hierachy

I m1 . 10−2 eV10−3 eV . |Mee| . 10−2 eV Excluded

III10−2 eV . m1 . 1 eV

10−2 eV . |Mee| . 1 eV|Ue3| & 3× 10−3

m1 . 10−1 eV|Mee| ' 5× 10−1 eV|Ue3| & 3× 10−2

IV10−2 eV . m1 . 1 eV

10−2 eV . |Mee| . 1 eV|Ue3| & 3× 10−3

m1 . 10−1 eV|Mee| ' 5× 10−1 eV|Ue3| & 3× 10−3

Table 6.3: Ansatz Predictions

82

Chapter 7

Concluding remarks

In the last thirty five years, Gauge theories have ruled the electroweak interactions with an enormoussuccess. The SM predicted new particles with a high precision. The existence of weak chargedcurrents and intermediate vector bosons, as well their mass relation was experimentally confirmed.

In the SM, the weak interactions are based on a local gauge symmetry group SU(2)L × U(1)Yspontaneously broken by the Higgs mechanism. The matter fields, leptons and quarks, are organizedin families (up to now three), being the left components weak isodoublets and the right componentsweak isosingltes. The vectorial bosons, W±µ , Z0

µ and γ are the interaction mediators, which can beproduce through annihilation or scattering processes. The scalar potential is the essential featureof the SM, it is added to the model in order to generate the gauge bosons and fermion masses, theessence of the Higgs mechanism. Moreover, a new scalar particle is predicted without being seen inthe actual colliders, a hope for the feature experiments.

After more than seventy years since birth of neutrino physics, neutrinos are still the most chal-lenging particles of intense investigation. This is related with the fact that they seem to behave verydifferently from the image stated by the SM. In spite of its success, there are many aspects in theSM which are not covered and are open questions, establishing it as an excellent effective theory. Asan ‘effective theory’ the SM is an excellent formalism to describe the low-energy phenomena, apartof the recent experiments on neutrino oscillations. The main reason for this is that the SM does notprovide an explanation for the disappearance of solar, atmospheric and reactor neutrinos. On theother hand, an effective theory can include effective interactions. If we allow this to happen, even atthe cost of giving-up some non-fundamental symmetry principle, then we are probably safe. But eventhough, we may ask why neutrinos should be so different from quarks and charged leptons. Here, themost reasonable answer seems to be closely related to very high energy physics.

The seesaw mechanism provides an elegant and economical explanation for the smallness of neu-trino masses in the sense that it only requires very heavy right handed neutrinos, not protected byany gauge symmetry. Moreover this phenomenon may be related with the breaking of a fundamentalsymmetry. Still, and spite of all the interesting features of this mechanism, the truth is that it is asreliable as any other which does not pose a conflict with the experimental data. Although this mayseem a very pessimistic point of view, one should always recall that this is how things usually evolvein particle physics.

In almost all experiments, neutrinos are produced through weak charged currents. If neutrinoshave mass, the neutrino produced is a superposition of different mass eigenstates. Therefore, throughthe evolution in time, the eigenstates will evolve and a different flavour state can be found in a later

83

time. This was considered by Pontecorvo in 1957 and is denoted by neutrino oscillation.The discovery of neutrino masses pushes to the study of the leptonic masses and their mixings.

It is a very challenging subject since we are forced to go beyond the SM, where mass matrices forneutrinos are not possible. Thus, a good understanding of the lepton mass matrices could give us anidea of symmetries that rule the world at very high energy scales. We have done a study for differentclasses of matrices based on texture zeros and we have predict some lower bounds for the lightestneutrino mass. Also predictions for the 2β0ν decay amplitude were made. It is important to enhancethat, mostly of the study in this area uses the basis where the charged leptons are the diagonal form.Thus, in such a basis, all the mass schemes with zeros in the position (1,1) have no contribution to2β0ν, this is indeed not the case for the parallel structures., the one we have considered in this thesis.

This texture zero analysis is still uncompleted, some hard work is need in order to look for usefulanalytical expressions. The analysis made is still relevant since it give us some physical quantitiespredictions. Moreover, this work done for the leptonic sector, can now be extended to the quarksector, a project for the near future.

The work presented in this thesis and all the open questions left behind are just a small sample ofhow work is still ahead in neutrino physics, not only on the theoretical but also on the experimentalside. In spite of the considerable number of neutrino experiments and all the efforts of theoreticalparticle physicists, there is a lot to be learnt from these particles. In the meanwhile, we will alwaysbe ready for more news about neutrinos...

84

Appendix A

Chirality Vs. helicity

There are usually a miss understanding between chirality and helicity. To understand those importantdefinitions let us start with the Dirac equation (iγµ∂µ −m)ψ = 0 and remember the definitions ofthe γµ matrices in the chiral representation

γ0 =

(0 I2

I2 0

), γi =

(0 σi

−σi 0

), γ5 =

(−I2 0

0 I2

)(A.1)

One can write the 4-component Dirac spinor in terms of 2-component Weyl spinors χ and η,

ψ =

(χ

η

), one can write the Dirac equation as

(0 i ∂∂t − i~σ.~∇

i ∂∂t + i~σ.~∇ 0

)(χ

η

)−m

(χ

η

)= 0 (A.2)

this gives two coupled equations. Using the plane wave solution

χ(x) = χ(p)e−ipµxµ , (A.3)

η(x) = η(p)e−ipµxµ , (A.4)

one obtain

Eχ+ ~σ.~pχ−mη = 0, (A.5)

Eη − ~σ.~pη −mχ = 0, (A.6)

these equations can be written as

~σ.~p

Eχ = −χ+

m

Eη

~σ.~p

Eη = η − m

Eχ. (A.7)

Now, knowing that the helicity operator is h = ~σ.~p|p| , one see that in the limit where p � m we

have E2 ' p2. This corresponds to two solutions, E =| p | (particle) and E = − | p | (antiparticle).One can then write Eq.(A.7) for positive energy as

85

hχ ' −χ+m

Eη (A.8)

hη ' η − m

Eχ (A.9)

and for negative energy as

hχ ' χ− m

Eη (A.10)

hη ' −η +m

Eχ (A.11)

This results must be interpreted as: In Eq.(A.8) we have a particle, E > 0, with negative helicity(left handed) and a fraction of the "wrong" helicity, positive helicity (right handed), suppressed bythe factor m/E. In Eq.(A.10) we have an antiparticle, E < 0, with positive helicity and a fraction ofnegative one, once again suppressed by the same factor. The same analysis can be done for Eq.(A.9)and Eq.(A.11), where one has a particle with almost positive helicity and an antiparticle with almostnegative helicity, respectively. Until now we have not talk about chirality, chiral spinor are defined as

PLψ =12

(1− γ5)ψ = ψL (A.12)

PRψ =12

(1 + γ5)ψ = ψR (A.13)

where the projectors satisfy

PL + PR = 1, P †L,R = PL,R, (A.14)

P 2L,R = PL,R, PL,RPR,L = 0 (A.15)

and also γ5ψR = ψR and γ5ψL = −ψL, in the limiting case of m = 0 chirality is the same ashelicity for particles and the opposite for antiparticles. To see that let us do the limit for E > 0, wehave

hψ =

(−1 0

0 1

)(χ

η

)= γ5ψ (A.16)

and for E < 0 one has

hψ =

(1 0

0 −1

)(χ

η

)= −γ5ψ (A.17)

In the representation of the γµ matrices that we give the projectors are given by

PL =12

(1− γ5) =

(I2 0

0 0

), PR =

12

(1 + γ5) =

(0 0

0 I2

)(A.18)

then, when we apply these projectors to the spinor field ψ we get PLψ = ψL =(χ 0

)Tand

PRψ = ψR =(

0 η)T

. Thus in the massless particle limit, this means that the projector PLsatisfy Eq.(A.8) and Eq.(A.10) so projects out a left handed particle (for E > 0) and a right handedantiparticle (for E < 0), and PR satisfy Eq.(A.9) and Eq.(A.11) thus it projects out a right handedparticle and a left handed antiparticle.

86

Appendix B

CP, T and CPT symmetries inneutrino oscillations.

We have sen that the operation C is not well defined for neutrinos since it converts a left handedneutrino into a non-existent left handed neutrino. The same for the operation P, which convertsa left handed neutrino into a non-existent right handed one. CP, T and CPT , however, are welldefined. One shall discuss this operations in the neutrino oscillations framework. Let us see what arethe conservation conditions of each operation when applied in a neutrino oscillation process

T : P (ναL → νβL; t) = P (νβL → ναL; t), (B.1)

CP : P (ναL → νβL; t) = P (ναR → νβR; t), (B.2)

CPT : P (ναL → νβL; t) = P (νβR → ναR; t). (B.3)

A direct result from these conditions is that, under the assumption of CPT invariance, if CP isconserved this is equivalent to T conservation. We shall now look with a little more detail the CPand CPT operations.

For the case of CP operation one should note that in quantum field theory, the same field operatorthat annihilates a particle also creates an antiparticle, whereas its hermitian conjugate does theopposite. Therefore, when we pass from neutrinos to antineutrinos oscillations the lepton mixingmatrix U amounts to U → U∗. This means that, if the mixing matrix in the leptonic sector iscomplex, from Eq.(5.3), CP is not conserved. To see the most general case of CP violation one shouldcount the physical phases for a general n × n mixing matrix. That was already done in subsection3.2.1 and section 4.4. We have for the case of Dirac neutrinos (n−1)(n−2)/2 physical phases, whichimplies n ≥ 3 generations to have CP violation. In the Majorana case we have n(n − 1)/2 physicalphases, out of these phases (n−1)(n−2)/2 are the usual Dirac-type phases while the remaining n−1

are the Majorana phases. Since the Majorana phases are some overall phases they do not lead to anyobservable, and thus do not affect neutrino oscillations. This means that in the Majorana case wealso have the condition n ≥ 3 for CP violation.

For the CPT transformation can be interpreted as the action of CP and the interchange betweenthe initial and final states. Therefore, this transformation sends U → U∗ and t→ −t, however, thisthus not affect the oscillation probability, Eq.(5.3). 1 In particular CPT invariance implies the samesurvival probabilities for neutrinos and antineutrinos.

1Remember that we are working in the ultrarelativistic limit where L ' t.

87

88

Bibliography

[1] A.Pich. The standard model of electroweak interactions. hep-ph/07054264, 2007.

[2] A.Strumia and F.Vissani. Neutrino masses and mixings and... hep-ph/0606054.

[3] G. C. Branco, L. Lavoura, and J. P. Silva. CP Violation. Oxford Science Publications, 1999.

[4] C.Csaki. Summer school on particle physics. In Higgs and Electroweak Symmetry Breaking, 2007.

[5] T.-P. Cheng and L.-F. Li. Gauge Theory of elementary particle physics. Oxford Science Publi-cations, 1996.

[6] P. data group. Particle physics, 2006.

[7] A. de Gouveia. Neutrino physics summer school. In Open Questions - The big Picture, 2007.

[8] E.Kh.Akhmedov. Neutrino physics. hep-ph/0001264, 2000.

[9] R. E.Marshak. Conseptual Foundations of Modern Particle Physics. World Scientific, 1993.

[10] B. et al. Astrophys.J., 496:505, 1998.

[11] M. et al. (CHOOZ Coll.). Initial results from the chooz long baseline reactor neutrino oscillationexperiment. Phys. Lett. B, 420:397, 1998. M. et al. (CHOOZ Coll.). Limits on neutrino oscilla-tions from the chooz experiment. Phys. Lett. B, 466:415, 1999. M. et al. (CHOOZ Coll.). Searchfor neutrino oscillations on a long base-line at the chooz nuclear power station. Eur.Phys.J.C,27:331, 2003.

[12] W. et al. (GALLEX Coll.). Gallex solar neutrino observations: results for gallex iv. Phys. Lett.B, 447:127, 1999.

[13] R.-S. et al. (IMB Coll.). Nucl.Phys.B (Proc. Suppl.), 38:331, 1995.

[14] L. K. Collaboration). Nucl.Phys.Proc. Suppl., 155:160, 2006. E. et al. (K2K Collaboration).Evidence for muon neutrino oscillation in an accelerator-based experiment. Phys. Rev. Lett.,94:081802, 2005. S. et al. ( K2K Coll.). Improved search for νµ → νe oscillation in a long-baseline accelerator experiment. Phys. Rev. Lett., 96:181801, 2006.

[15] K. et al. (Kamiokande Coll.). Solar neutrino data covering solar cycle 22. Phys. Rev. Lett.,77:1683, 1996.

[16] Y. F. et al. (Kamiokande Coll.). Atmospheric νµ/νe ratio in the multi-gev energy range. PhysicsLetters B, 335:237, 1994.

89

[17] K. et al. (KamLAND Coll.). First results from kamland: Evidence for reactor antineutrinodisappearance. Phys. Rev. Lett., 90:021802, 2003. T. et al. Measurement of neutrino oscillationwith kamland: Evidence of spectral distortion. Phys. Rev. Lett., 94:081801, 2005.

[18] B. et al. (KARMEN Coll.). Upper limits for neutrino oscillations νµ → nue from muon decayat rest. Phys. Rev. D, 65:112001, 2002. G.Drexlin. Nucl.Phys.Proc. Suppl., 118:164, 2006.

[19] C. et al. (LSND Coll.). Evidence for νµ → νe oscillations from the lsnd experiment at the losalamos meson physics facility. Phys. Rev. Lett., 77:3082, 1996. C. et al. (LSND Coll.). Results onνµ → νe neutrino oscillations from the lsnd experiment. Phys. Rev. Lett., 81:1774, 1998. C. et al.(LSND Coll.). Results on νµ → νe oscillations from pion decay in flight neutrinos. Phys. Rev.C, 58:2489, 1998.

[20] M. et al. (MACRO Collaboration). Measurements of atmospheric muon neutrino oscillations,global analysis of the data collected with macro detector. Eur.Phys.J.C, 36:323, 2004.

[21] D. et al. (MINOS Coll.). report NUMI-L-63, 1995.

[22] J. et al. (SAGE Coll.). The russian-american gallium experiment (sage) cr neutrino sourcemeasurement. Phys. Rev. Lett., 77:4708, 1996. J. et al. (SAGE Coll.). Measurement of the solarneutrino capture rate with gallium metal. Phys. Rev. C, 60:055801, 1996.

[23] Q. et al. (SNO Coll.). Measurement of the rate of νe + d → p + p + e− interactions producedby 8b solar neutrinos at the sudbury neutrino observatory. Phys. Rev. Lett., 87:071301, 2001.Q. et al. (SNO Coll.). Direct evidence for neutrino flavor transformation from neutral-currentinteractions in the sudbury neutrino observatory. Phys. Rev. Lett., 89:011301, 2002. Q. et al.(SNO Coll.). Measurement of day and night neutrino energy spectra at sno and constraints onneutrino mixing parameters. Phys. Rev. Lett., 89(011302), 2002. B. et al. (SNO Col.). Electronenergy spectra, fluxes, and day-night asymmetries of 8b solar neutrinos from measurements withnacl dissolved in the heavy-water detector at the sudbury neutrino observatory. Phys. Rev. C,72:05502, 2005.

[24] M. et al. (Soudan 2 Collaboration). Measurement of the l/e distributions of atmospheric ν insoudan 2 and their interpretation as neutrino oscillations. Phys. Rev. D, 68:113004, 2003.

[25] Y. F. et al. (Super-Kamiokande Coll.). Measurements of the solar neutrino flux fromsuper-kamiokande’s first 300 days. Phys. Rev. Lett., 81:1158, 1998. Y. F. et al. (Super-Kamiokande Coll.). Measurement of the solar neutrino energy spectrum using neutrino-electronscattering. Phys. Rev. Lett., 82:2430, 1999. M. S.-K. Coll.). Solar neutrinos with super-kamiokande. hep-ex/9903034. S. et al. (Super-Kamiokande Coll.). Solar 8b and hep neu-trino measurements from 1258 days of super-kamiokande data. Phys. Rev. Lett., 86:5651,2001. S. et al. (Super-Kamiokande Coll.). Constraints on neutrino oscillations using 1258 daysof super-kamiokande solar neutrino data. Phys. Rev. Lett., 86:5656, 2001. S. et al. (Super-Kamiokande Coll.). Determination of solar neutrino oscillation parameters using 1496 days ofsuper-kamiokande-i data. Phys. Lett. B, 539:179, 2002. M. et al. (Super-Kamiokande Coll.). Pre-cise measurement of the solar neutrino day-night and seasonal variation in super-kamiokande-i.Phys. Rev. D, 69:011104, 2004.

90

[26] P. H. Frampton, S. L. Glashow, and D. Marfatia. Zeroes of neutrino mass matrix. PhysicsLetters B, 2002.

[27] G.Drexlin. Nucl.Phys.Proc. Suppl., 118:164, 2006.

[28] C. Giunti. Flavor neutrinos states. hep-ph/0402217, 2004.

[29] I.Stancu. Nucl.Phys.Proc. Suppl., 155:164, 2006.

[30] B. Kayser. Neutrino physics summer school. In Neutrino Phenomenology, 2007.

[31] M.Maltoni, T.Schwetz, M.A.Tortola, and J.W.F.Valle. Status of global fits to neutrino oscilla-tions. New J.Phys., 6:122, 2004.

[32] Y. Nir. Cp violation in meson decays. hep-ph/0510413V1, 2005.

[33] P. Sikivie. The pooltable analogy to axion physics. hep-ph/9506229v1, 1995.

[34] S.M.Bilenky, C.Giunti, and C.W.Kim. Finally neutrino has mass. hep-ph/9902462.

[35] S. T.Morii, C.S.Lim. The Physics of the Standard Model and Beyond. World Scientific, 2004.

[36] W.E.Burcham and M.Jobes. Nuclear and Particle Physics. Longman Scientific and Technical,1995.

[37] G.C. Branco, D. Emmanuel-Costa and R. Gonzalez Felipe Texture zeros and weak basis trans-formations. Phys.Lett. B, 477:147, 2000

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