# Some Aspects of Chiral Perturbation Theory and Neutrino

## Transcript Of Some Aspects of Chiral Perturbation Theory and Neutrino

Some Aspects of Chiral Perturbation Theory and Neutrino Physics

Tesis Doctoral

Doctorado en Física

Mayo 2017

Mehran Zahiri Abyaneh

IFIC - Universitat de València - CSIC Departament de Física Teòrica

Director: Antonio Pich Zardoya

To My Mother and Deceased Father

Antonio Pich Zardoya, catedrático de la Universidad de Valencia, CERTIFICA: Que la presente memoria "Some Aspects of Chiral Perturbation Theory and Neutrino Physics" ha sido realizada bajo su dirección en el Instituto de Física Corpuscular, centro mixto de la Universidad de Valencia y del CSIC, por Mehran Zahiri Abyaneh y constituye su Tesis para optar al grado de Doctor en Física. Y para que así conste, en cumplimiento de la legislación vigente, presenta en el Departamento de Física Teórica de la Universidad de Valencia la referida Tesis Doctoral, y ﬁrma el presente certiﬁcado.

Valencia, a 30 de Mayo de 2017.

Antonio Pich Zardoya

Contents

Introduction

1

1 The Standard Model of Particle Physics

6

1.1 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Shortcomings of the SM . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Eﬀective Field Theories and Chiral Perturbation Theory

14

2.1 Matching in Eﬀective Field Theories . . . . . . . . . . . . . . . . . 16

2.2 QCD and the Chiral Lagrangian . . . . . . . . . . . . . . . . . . . 21

2.3 External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Non-Linear Sigma Model . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 ChPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 The First paper (see Chapter 4) . . . . . . . . . . . . . . . . . . . 31

2.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Neutrino physics and Renormalization Group Equations

43

3.1 Limits on the Neutrino masses . . . . . . . . . . . . . . . . . . . . 44

3.2 Neutrino mass and See-saw mechanism . . . . . . . . . . . . . . . . 45

3.2.1 The See-saw mechanism . . . . . . . . . . . . . . . . . . . . 46

3.3 Neutrino oscillations in vacuum . . . . . . . . . . . . . . . . . . . . 50

3.3.1 Two ﬂavor case . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.2 CP and T violating eﬀects . . . . . . . . . . . . . . . . . . . 55

3.4 The HSMU Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Minimal Supersymmetric Model . . . . . . . . . . . . . . . . . . . 58

3.6 Renormalization Group Equations . . . . . . . . . . . . . . . . . . 59

3.7 Running neutrino mass parameters . . . . . . . . . . . . . . . . . . 62

3.8 The Second Paper (see Chapter 5) . . . . . . . . . . . . . . . . . . 65

3.8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

ii

Contents

iii

3.8.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.8.3 Eﬀects of the Large tan β and Threshold Corrections . . . . 66 3.8.4 Diﬀerent Scenarios for the Proportionality at High Scale . . 68 3.8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.9 The Third Paper (see Chapter 6) . . . . . . . . . . . . . . . . . . . 72 3.9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 On the Minimality of the Order p6 Chiral Lagrangian

81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2 Chiral perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Outline of the method . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4 SU(2) case with s = p = 0 . . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 High Scale Mixing Relations as a Natural Explanation for Large

Neutrino Mixing

98

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 RG evolution of the leptonic mixing parameters . . . . . . . . . . . 103

5.3 The low energy SUSY threshold corrections and the absolute neu-

trino mass scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3.1 The low energy SUSY threshold corrections . . . . . . . . . 107

5.3.2 The absolute neutrino mass scale . . . . . . . . . . . . . . . 108

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.1 RG evolution of HSMR . . . . . . . . . . . . . . . . . . . . 110

5.4.2 Phenomenology of HSMR . . . . . . . . . . . . . . . . . . . 112

5.5 Theoretical models for high scale mixing relations . . . . . . . . . . 133

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6 Precise Predictions For Dirac Neutrino mixing

143

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 RG evolution of the neutrino mixing parameters for Dirac neutrinos145

6.2.1 Results for the SUSY breaking scale at 2 TeV . . . . . . . . 146

6.2.2 Variation of the SUSY breaking scale . . . . . . . . . . . . 149

6.2.3 Variation of the uniﬁcation scale . . . . . . . . . . . . . . . 150

6.3 Model for the HSMR parametrization . . . . . . . . . . . . . . . . 151

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

iv

Contents

Conclusions

159

7 Resumen de la Tesis

163

7.1 Física de ChPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.1.1 Objetivos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.1.2 Metodología . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.1.3 Resultados del Primer Artículo (el Capítulo 4) . . . . . . . 167

7.2 Física de los Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2.1 Objetivos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2.2 Metodología . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.2.3 Resultados del Segundo Artículo (el Capítulo 5) . . . . . . 173

7.2.4 Resultados del Tercer Artículo (el Capítulo 6) . . . . . . . 174

Acknowledgements

178

Introduction

The advent of Spontaneous Symmetry Breaking (SSB) by Yoichiro Nambu [1]– [3] from Superconductivity to the particle physics community was the beginning of an era, whose consequences are still fruitful. SSB is a phenomenon where a symmetry in the basic laws of physics appears to be broken. For example, when an standing straight rod which has the rotational symmetry, that is, it looks the same from any horizontal direction, is pressurized from the top, it will bend in some direction, and the rotational symmetry is lost. Since all directions are equivalently probable to be chosen for bending, one says the symmetry is broken spontaneously. There are many examples of SSB in Quantum mechanics and solid state physics. For example, ferro-magnets, rotational invariance in crystals, etc.

In the language of Quantum Field Theory (QFT), a system is said to possess a symmetry that is spontaneously broken if the Lagrangian describing the dynamics of the system is invariant under these symmetry transformations, but the vacuum is not [4]. Since, the vacuum has many intrinsic degrees of freedom, SSB can play an important role and as the universe expands and cools down, several SSB phase transitions from states of higher symmetries to lower ones might have happened. In fact, this is the ultimate dream of the particle physics community to realize, what was the original symmetry of nature, before any SSB took place.

The SSB can happen in two ways that is, the symmetry which is broken can be global or local, which has completely diﬀerent consequences. We will here describe brieﬂy both types of SSB and then will consider some examples of each in coming chapters. Lectures about the phenomenon of spontaneous symmetry breaking of a global symmetry and how the situation changes in the presence of a local gauge symmetry can be found in refs. [5, 6].

First a recount of the history as Nambu describes it himself [1], One day before publication of the BCS paper, Bob Schrieﬀer, still a student, came to Chicago to give a seminar on the BCS theory in progress . . . I was very much disturbed by the fact that their wave function did not conserve electron number. It did not make sense . . . At the same time I was impressed by their boldness and tried to understand the problem. So, the main reason which led him to the idea was the fact that, as it turns out, in the BCS model of superconductivity [7], the quasi

1

2

Introduction

particles introduced by Bogoliubov [8] and Valatin [9](BV), which are the building blocks of the Cooper pairs, seem not to have a deﬁnite charge. This means that the electric charge is not conserved which leads to problems for electromagnetic phenomena like the Meissner eﬀect. Therefore, he introduced the notion of a massless spin–zero collective mode; to be called later on the Nambu–Goldstone (NG) boson; that appears due to the spontaneously broken continuous gauge symmetry and rescues the charge conservation [2]. This is an example of the SSB for a local symmetry or as Weinberg puts it [10], A superconductor is simply a material in which electromagnetic gauge invariance is spontaneously broken.

Soon after the introduction of the notion of the spontaneously broken continuous local symmetry in superconductors, due to the similarity of the BV equation to the Dirac equation, Nambu and Jona-Lasinio (NJL) transported the BCS theory to nuclear physics [3]. In this case, the axial symmetry as an approximately conserved global symmetry in ﬂavor space, is spontaneously broken. Therefore, the nucleon mass is generated by an SSB of chirality, and the pion is the NG boson of this symmetry breaking. In the limit of exact conservation, the pion will become massless and the proton and the neutron masses will also become the same.

On the other hand, in 1962 Goldstone showed [11] that spontaneous breaking of a global symmetry in a relativistic ﬁeld theory results in massless spin-zero bosons. According to the Goldstone theorem: if a theory has a global symmetry of the Lagrangian, which is not a symmetry of the vacuum, then there must exist one massless boson, scalar or pseudoscalar, associated to each generator which does not annihilate the vacuum and having its same quantum numbers. These modes are referred to as Nambu-Goldstone bosons or simply as Goldstone bosons.

So, the NJL model is an example of SSB, where the Goldstone theorem applies. It was the ﬁrst model to introduce pion as a Nambu-Goldstone boson of the broken chiral symmetry in QCD, but not the last one. In fact, it suﬀers from lack of conﬁnement and is nonrenormalizable in four space–time dimensions. Therefore, this model is regarded as an eﬀective theory for the QCD, which needs to be UV completed. There are other eﬀective theories to describe dynamics of mesons like Chiral perturbation Theory (ChPT), which we will discuss in detail later on, after introducing the notion of eﬀective ﬁeld theories in general. We will also discuss a work related to ChPT.

Back to the history, after Goldstone’s prediction of massless modes, the problem was that apart from the pions in nuclear physics, they were excluded experimentally in QFT and therefore, at the time the application of SSB to the QFT was not clear. In fact, solution to this problem also came from the solid state

Introduction

3

community. The year before Goldstone published his paper, Philip Anderson had pointed out [12] that, in a superconductor where the local gauge symmetry is broken spontaneously, the Goldstone (plasmon) mode becomes massive due to the gauge ﬁeld interaction and is eﬀectively eaten by the photon to become a ﬁnite-mass longitudinal mode (Meissner eﬀect), despite the gauge invariance. But, he did not discuss any relativistic model and so, since Lorentz invariance was a crucial ingredient of the Goldstone theorem, he did not demonstrate that NG modes could be evaded.

Finally, following the work of Goldstone, Anderson and Nambu, in 1964 realistic models with Lorentz invariance and non-Abelian gauge ﬁelds were formulated by Higgs and others [13,14]. They showed that in the case when a gauge symmetry is broken spontaneously, the Goldstone’s theorem does not apply and another mechanism comes to rescue, the so-called Higgs mechanism [13]. The would-be Goldstone bosons associated to the global symmetry breaking do not manifest explicitly in the physical spectrum but instead they combine with the massless gauge bosons and as a result, once the spectrum of the theory is built up on the asymmetrical vacuum, there appear massive vector particles. The number of vector bosons that acquire a mass is precisely equal to the number of these would-be-Goldstone bosons. This led Glashow–Weinberg–Salam (GWS) [15–17] to develop the electroweak theory as a part of the Standard Model of particle physics (SM). We will describe the SM of Electroweak (EW) interactions in the next chapter and will present two works related to the Higgs mechanism later on, in the framework of the neutrino physics.

As a side note, Higgs also predicted that due to this SSB a new scalar mode will appear in the particle spectrum of the theory, nowadays known as the Higgs boson. This was ﬁnally detected in 2012 in the Large Hadron Collider at CERN. But ironically, the only Higgs boson to be discovered experimentally before 2012 was also detected in solid state physics as an unexpected feature of the Raman spectrum of N bSe2, an oscillation of the amplitude of the superconducting gap [18],

The outline of this thesis is the following. In chapter 1 we will discuss the Standard Model of electroweak interactions, which is also relevant to neutrino physics. The chapter 2 brieﬂy introduces the notion of eﬀective ﬁeld theories and discusses symmetries of the QCD Lagrangian in the ﬂavor space. Afterwards, it introduces the ChPT as an eﬀective ﬁeld theory. A prologue to the paper on ChPT is given at the end of this chapter. In chapter 3, using the information from previous chapters, the neutrino physics is considered, where the Renormalization Group (RG) equations for neutrino parameters are also discussed. A prologue to

4

Introduction

the two papers, related to neutrino physics, is given at the end of this chapter. The three papers constituting the bulk of the thesis are presented subsequently.

Tesis Doctoral

Doctorado en Física

Mayo 2017

Mehran Zahiri Abyaneh

IFIC - Universitat de València - CSIC Departament de Física Teòrica

Director: Antonio Pich Zardoya

To My Mother and Deceased Father

Antonio Pich Zardoya, catedrático de la Universidad de Valencia, CERTIFICA: Que la presente memoria "Some Aspects of Chiral Perturbation Theory and Neutrino Physics" ha sido realizada bajo su dirección en el Instituto de Física Corpuscular, centro mixto de la Universidad de Valencia y del CSIC, por Mehran Zahiri Abyaneh y constituye su Tesis para optar al grado de Doctor en Física. Y para que así conste, en cumplimiento de la legislación vigente, presenta en el Departamento de Física Teórica de la Universidad de Valencia la referida Tesis Doctoral, y ﬁrma el presente certiﬁcado.

Valencia, a 30 de Mayo de 2017.

Antonio Pich Zardoya

Contents

Introduction

1

1 The Standard Model of Particle Physics

6

1.1 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Shortcomings of the SM . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Eﬀective Field Theories and Chiral Perturbation Theory

14

2.1 Matching in Eﬀective Field Theories . . . . . . . . . . . . . . . . . 16

2.2 QCD and the Chiral Lagrangian . . . . . . . . . . . . . . . . . . . 21

2.3 External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Non-Linear Sigma Model . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 ChPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 The First paper (see Chapter 4) . . . . . . . . . . . . . . . . . . . 31

2.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Neutrino physics and Renormalization Group Equations

43

3.1 Limits on the Neutrino masses . . . . . . . . . . . . . . . . . . . . 44

3.2 Neutrino mass and See-saw mechanism . . . . . . . . . . . . . . . . 45

3.2.1 The See-saw mechanism . . . . . . . . . . . . . . . . . . . . 46

3.3 Neutrino oscillations in vacuum . . . . . . . . . . . . . . . . . . . . 50

3.3.1 Two ﬂavor case . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.2 CP and T violating eﬀects . . . . . . . . . . . . . . . . . . . 55

3.4 The HSMU Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Minimal Supersymmetric Model . . . . . . . . . . . . . . . . . . . 58

3.6 Renormalization Group Equations . . . . . . . . . . . . . . . . . . 59

3.7 Running neutrino mass parameters . . . . . . . . . . . . . . . . . . 62

3.8 The Second Paper (see Chapter 5) . . . . . . . . . . . . . . . . . . 65

3.8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

ii

Contents

iii

3.8.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.8.3 Eﬀects of the Large tan β and Threshold Corrections . . . . 66 3.8.4 Diﬀerent Scenarios for the Proportionality at High Scale . . 68 3.8.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.9 The Third Paper (see Chapter 6) . . . . . . . . . . . . . . . . . . . 72 3.9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9.2 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 On the Minimality of the Order p6 Chiral Lagrangian

81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2 Chiral perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Outline of the method . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4 SU(2) case with s = p = 0 . . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 High Scale Mixing Relations as a Natural Explanation for Large

Neutrino Mixing

98

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2 RG evolution of the leptonic mixing parameters . . . . . . . . . . . 103

5.3 The low energy SUSY threshold corrections and the absolute neu-

trino mass scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3.1 The low energy SUSY threshold corrections . . . . . . . . . 107

5.3.2 The absolute neutrino mass scale . . . . . . . . . . . . . . . 108

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.1 RG evolution of HSMR . . . . . . . . . . . . . . . . . . . . 110

5.4.2 Phenomenology of HSMR . . . . . . . . . . . . . . . . . . . 112

5.5 Theoretical models for high scale mixing relations . . . . . . . . . . 133

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6 Precise Predictions For Dirac Neutrino mixing

143

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 RG evolution of the neutrino mixing parameters for Dirac neutrinos145

6.2.1 Results for the SUSY breaking scale at 2 TeV . . . . . . . . 146

6.2.2 Variation of the SUSY breaking scale . . . . . . . . . . . . 149

6.2.3 Variation of the uniﬁcation scale . . . . . . . . . . . . . . . 150

6.3 Model for the HSMR parametrization . . . . . . . . . . . . . . . . 151

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

iv

Contents

Conclusions

159

7 Resumen de la Tesis

163

7.1 Física de ChPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.1.1 Objetivos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.1.2 Metodología . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.1.3 Resultados del Primer Artículo (el Capítulo 4) . . . . . . . 167

7.2 Física de los Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2.1 Objetivos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2.2 Metodología . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.2.3 Resultados del Segundo Artículo (el Capítulo 5) . . . . . . 173

7.2.4 Resultados del Tercer Artículo (el Capítulo 6) . . . . . . . 174

Acknowledgements

178

Introduction

The advent of Spontaneous Symmetry Breaking (SSB) by Yoichiro Nambu [1]– [3] from Superconductivity to the particle physics community was the beginning of an era, whose consequences are still fruitful. SSB is a phenomenon where a symmetry in the basic laws of physics appears to be broken. For example, when an standing straight rod which has the rotational symmetry, that is, it looks the same from any horizontal direction, is pressurized from the top, it will bend in some direction, and the rotational symmetry is lost. Since all directions are equivalently probable to be chosen for bending, one says the symmetry is broken spontaneously. There are many examples of SSB in Quantum mechanics and solid state physics. For example, ferro-magnets, rotational invariance in crystals, etc.

In the language of Quantum Field Theory (QFT), a system is said to possess a symmetry that is spontaneously broken if the Lagrangian describing the dynamics of the system is invariant under these symmetry transformations, but the vacuum is not [4]. Since, the vacuum has many intrinsic degrees of freedom, SSB can play an important role and as the universe expands and cools down, several SSB phase transitions from states of higher symmetries to lower ones might have happened. In fact, this is the ultimate dream of the particle physics community to realize, what was the original symmetry of nature, before any SSB took place.

The SSB can happen in two ways that is, the symmetry which is broken can be global or local, which has completely diﬀerent consequences. We will here describe brieﬂy both types of SSB and then will consider some examples of each in coming chapters. Lectures about the phenomenon of spontaneous symmetry breaking of a global symmetry and how the situation changes in the presence of a local gauge symmetry can be found in refs. [5, 6].

First a recount of the history as Nambu describes it himself [1], One day before publication of the BCS paper, Bob Schrieﬀer, still a student, came to Chicago to give a seminar on the BCS theory in progress . . . I was very much disturbed by the fact that their wave function did not conserve electron number. It did not make sense . . . At the same time I was impressed by their boldness and tried to understand the problem. So, the main reason which led him to the idea was the fact that, as it turns out, in the BCS model of superconductivity [7], the quasi

1

2

Introduction

particles introduced by Bogoliubov [8] and Valatin [9](BV), which are the building blocks of the Cooper pairs, seem not to have a deﬁnite charge. This means that the electric charge is not conserved which leads to problems for electromagnetic phenomena like the Meissner eﬀect. Therefore, he introduced the notion of a massless spin–zero collective mode; to be called later on the Nambu–Goldstone (NG) boson; that appears due to the spontaneously broken continuous gauge symmetry and rescues the charge conservation [2]. This is an example of the SSB for a local symmetry or as Weinberg puts it [10], A superconductor is simply a material in which electromagnetic gauge invariance is spontaneously broken.

Soon after the introduction of the notion of the spontaneously broken continuous local symmetry in superconductors, due to the similarity of the BV equation to the Dirac equation, Nambu and Jona-Lasinio (NJL) transported the BCS theory to nuclear physics [3]. In this case, the axial symmetry as an approximately conserved global symmetry in ﬂavor space, is spontaneously broken. Therefore, the nucleon mass is generated by an SSB of chirality, and the pion is the NG boson of this symmetry breaking. In the limit of exact conservation, the pion will become massless and the proton and the neutron masses will also become the same.

On the other hand, in 1962 Goldstone showed [11] that spontaneous breaking of a global symmetry in a relativistic ﬁeld theory results in massless spin-zero bosons. According to the Goldstone theorem: if a theory has a global symmetry of the Lagrangian, which is not a symmetry of the vacuum, then there must exist one massless boson, scalar or pseudoscalar, associated to each generator which does not annihilate the vacuum and having its same quantum numbers. These modes are referred to as Nambu-Goldstone bosons or simply as Goldstone bosons.

So, the NJL model is an example of SSB, where the Goldstone theorem applies. It was the ﬁrst model to introduce pion as a Nambu-Goldstone boson of the broken chiral symmetry in QCD, but not the last one. In fact, it suﬀers from lack of conﬁnement and is nonrenormalizable in four space–time dimensions. Therefore, this model is regarded as an eﬀective theory for the QCD, which needs to be UV completed. There are other eﬀective theories to describe dynamics of mesons like Chiral perturbation Theory (ChPT), which we will discuss in detail later on, after introducing the notion of eﬀective ﬁeld theories in general. We will also discuss a work related to ChPT.

Back to the history, after Goldstone’s prediction of massless modes, the problem was that apart from the pions in nuclear physics, they were excluded experimentally in QFT and therefore, at the time the application of SSB to the QFT was not clear. In fact, solution to this problem also came from the solid state

Introduction

3

community. The year before Goldstone published his paper, Philip Anderson had pointed out [12] that, in a superconductor where the local gauge symmetry is broken spontaneously, the Goldstone (plasmon) mode becomes massive due to the gauge ﬁeld interaction and is eﬀectively eaten by the photon to become a ﬁnite-mass longitudinal mode (Meissner eﬀect), despite the gauge invariance. But, he did not discuss any relativistic model and so, since Lorentz invariance was a crucial ingredient of the Goldstone theorem, he did not demonstrate that NG modes could be evaded.

Finally, following the work of Goldstone, Anderson and Nambu, in 1964 realistic models with Lorentz invariance and non-Abelian gauge ﬁelds were formulated by Higgs and others [13,14]. They showed that in the case when a gauge symmetry is broken spontaneously, the Goldstone’s theorem does not apply and another mechanism comes to rescue, the so-called Higgs mechanism [13]. The would-be Goldstone bosons associated to the global symmetry breaking do not manifest explicitly in the physical spectrum but instead they combine with the massless gauge bosons and as a result, once the spectrum of the theory is built up on the asymmetrical vacuum, there appear massive vector particles. The number of vector bosons that acquire a mass is precisely equal to the number of these would-be-Goldstone bosons. This led Glashow–Weinberg–Salam (GWS) [15–17] to develop the electroweak theory as a part of the Standard Model of particle physics (SM). We will describe the SM of Electroweak (EW) interactions in the next chapter and will present two works related to the Higgs mechanism later on, in the framework of the neutrino physics.

As a side note, Higgs also predicted that due to this SSB a new scalar mode will appear in the particle spectrum of the theory, nowadays known as the Higgs boson. This was ﬁnally detected in 2012 in the Large Hadron Collider at CERN. But ironically, the only Higgs boson to be discovered experimentally before 2012 was also detected in solid state physics as an unexpected feature of the Raman spectrum of N bSe2, an oscillation of the amplitude of the superconducting gap [18],

The outline of this thesis is the following. In chapter 1 we will discuss the Standard Model of electroweak interactions, which is also relevant to neutrino physics. The chapter 2 brieﬂy introduces the notion of eﬀective ﬁeld theories and discusses symmetries of the QCD Lagrangian in the ﬂavor space. Afterwards, it introduces the ChPT as an eﬀective ﬁeld theory. A prologue to the paper on ChPT is given at the end of this chapter. In chapter 3, using the information from previous chapters, the neutrino physics is considered, where the Renormalization Group (RG) equations for neutrino parameters are also discussed. A prologue to

4

Introduction

the two papers, related to neutrino physics, is given at the end of this chapter. The three papers constituting the bulk of the thesis are presented subsequently.