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Contents

Preface xiii

1 Physics of Massive Neutrinos 1Anjan S. Joshipura

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 11.1.1 History . . . . . . . . . . . . . . . . . . . . 11.1.2 Fermi theory and beyond . . . . . . . . . . 3

1.2 Neutrino masses: basic formalism . . . . . . . . . . 71.2.1 Neutrino mass terms . . . . . . . . . . . . . 91.2.2 Dirac neutrino . . . . . . . . . . . . . . . . 131.2.3 Pseudo-Dirac neutrino . . . . . . . . . . . . 131.2.4 Seesaw masses . . . . . . . . . . . . . . . . 14

1.3 Neutrino mass measurements: direct detection . . 151.3.1 Beta decay and m e . . . . . . . . . . . . . 161.3.2 Neutrinoless double beta decay . . . . . . . 17

1.4 Neutrino mass measurements: oscillations . . . . . 191.4.1 Atmospheric neutrinos . . . . . . . . . . . . 211.4.2 Solar neutrinos . . . . . . . . . . . . . . . . 231.4.3 Mechanisms for solar neutrino conversion . 251.4.4 Vacuum oscillations . . . . . . . . . . . . . 251.4.5 MSW mechanism . . . . . . . . . . . . . . . 26

1.5 Neutrino mass spectrum from experiments . . . . . 291.6 Models of neutrino masses . . . . . . . . . . . . . . 31

1.6.1 Seesaw models . . . . . . . . . . . . . . . . 321.6.2 Radiative models . . . . . . . . . . . . . . . 33

1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . 35

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viii CONTENTS

2 Higgs Physics 39Saurabh D. Rindani

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 392.2 The Standard Model Higgs boson . . . . . . . . . . 40

2.2.1 Introduction to the Standard Model . . . . 402.2.2 Spontaneous symmetry breaking . . . . . . 432.2.3 Properties of the Standard Model Higgs boson 472.2.4 Interactions with fermions . . . . . . . . . . 502.2.5 Summary of the properties of SM Higgs . . 52

2.3 Theoretical constraints on the mass of the SM Higgs 522.3.1 Bound from vacuum stability . . . . . . . . 522.3.2 Upper bound on the Higgs mass . . . . . . 542.3.3 Unitarity limit . . . . . . . . . . . . . . . . 562.3.4 W L W L scattering . . . . . . . . . . . . . . . 59

2.4 Experimental constraints on the mass of the SMHiggs . . . . . . . . . . . . . . . . . . . . . . . . . 612.4.1 Electroweak precision data . . . . . . . . . 61

2.5 Decays of the SM Higgs boson . . . . . . . . . . . 622.5.1 Higgs decays into fermions . . . . . . . . . . 622.5.2 Higgs decays into W W and ZZ . . . . . . . 632.5.3 Higgs decays into gluon and photon pairs . 642.5.4 Higgs decay into Z . . . . . . . . . . . . . 662.5.5 Conclusion . . . . . . . . . . . . . . . . . . 66

2.6 Production of Higgs at e+ ecolliders . . . . . . . . 672.6.1 Bjorken process . . . . . . . . . . . . . . . . 682.6.2 Associated Bjorken process . . . . . . . . . 732.6.3 Gauge-boson fusion process . . . . . . . . . 732.6.4 Photon-photon fusion . . . . . . . . . . . . 752.6.5 Higgs production at hadron colliders . . . . 762.6.6 Gluon-gluon fusion . . . . . . . . . . . . . . 772.6.7 Vector boson fusion . . . . . . . . . . . . . 802.6.8 Bjorken-like process . . . . . . . . . . . . . 822.6.9 Higgs bremsstrahlung off top quark . . . . . 82

2.7 Present status and outlook . . . . . . . . . . . . . 82

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CONTENTS ix

3 CP-violation: A Pedagogical Introduction 89Biswarup Mukhopadhyaya

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 893.2 Parity, charge conjugation and time reversal . . . . 923.2.1 Parity P . . . . . . . . . . . . . . . . . . . . 923.2.2 Charge conjugation C . . . . . . . . . . . . 973.2.3 Time reversal T . . . . . . . . . . . . . . . 983.2.4 Basic principle ( P , C or T ) . . . . . . . . . 100

3.3 A complex phase causes CP-violation . . . . . . . . 1023.4 CP-violation in the Standard Model . . . . . . . . 1033.5 The unitarity triangles . . . . . . . . . . . . . . . . 1103.6 The neutral kaon system . . . . . . . . . . . . . . . 1113.7 CP-violation in neutral kaons: observables . . . . . 113

3.8 CP-violating parameters from the Standard Model 1193.8.1 The calculation of . . . . . . . . . . . . . 1193.8.2 Evaluation of I . . . . . . . . . . . . . . . . 1233.8.3 / in the Standard Model . . . . . . . . . 126

3.9 CP-violation in B -meson decays . . . . . . . . . . . 1273.9.1 Charged B -decays . . . . . . . . . . . . . . 1293.9.2 Neutral B -decay: cleaner grounds . . . . . 130

4 Lectures on Perturbative Quantum Chromodynam-ics 137Compiled by Debashis Ghoshal & V. Ravindran

4.1 Lecture I . . . . . . . . . . . . . . . . . . . . . . . 1374.2 Lecture II . . . . . . . . . . . . . . . . . . . . . . . 1424.3 Lecture III . . . . . . . . . . . . . . . . . . . . . . 1474.4 Lecture IV . . . . . . . . . . . . . . . . . . . . . . . 1554.5 Lecture V . . . . . . . . . . . . . . . . . . . . . . . 1614.6 Lecture VI . . . . . . . . . . . . . . . . . . . . . . . 1654.7 Lecture VII . . . . . . . . . . . . . . . . . . . . . . 1724.8 Lecture VIII . . . . . . . . . . . . . . . . . . . . . . 1804.9 Lecture IX . . . . . . . . . . . . . . . . . . . . . . . 1884.10 Lecture X . . . . . . . . . . . . . . . . . . . . . . . 196

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4.11 Lecture XI . . . . . . . . . . . . . . . . . . . . . . . 203

5 Quark-Gluon Plasma 209

Rajiv V. Gavai

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2095.1.1 Connement & why quark-gluon plasma is

important . . . . . . . . . . . . . . . . . . . 2105.1.2 A simple model for quark-gluon plasma: the

MIT bag model . . . . . . . . . . . . . . . . 2125.1.3 -T phase diagram . . . . . . . . . . . . . . 213

5.2 QGP from QCD . . . . . . . . . . . . . . . . . . . 2215.2.1 Path integral formalism . . . . . . . . . . . 2215.2.2 Spectral function . . . . . . . . . . . . . . . 225

5.2.3 Neutral scalar elds . . . . . . . . . . . . . 2275.2.4 Infra-red problems . . . . . . . . . . . . . . 232

5.3 QGP from QCD II . . . . . . . . . . . . . . . . . . 2365.3.1 Quark and gluon elds . . . . . . . . . . . . 2365.3.2 Symmetries and order parameters . . . . . 2445.3.3 Free energy in pertubation theory . . . . . 2485.3.4 Hierarchy of scales in QCD at nite temper-

ature . . . . . . . . . . . . . . . . . . . . . . 2505.4 QGP and lattice QCD . . . . . . . . . . . . . . . . 252

5.4.1 Lattice fermions . . . . . . . . . . . . . . . 2535.4.2 Gauge elds . . . . . . . . . . . . . . . . . . 2565.4.3 Calculational techniques . . . . . . . . . . . 2595.4.4 Continuum limit and renormalization . . . 2615.4.5 Physical observables . . . . . . . . . . . . . 266

5.5 Heavy ion collisions . . . . . . . . . . . . . . . . . . 2705.6 Signals of QGP . . . . . . . . . . . . . . . . . . . . 278

5.6.1 J/ suppression . . . . . . . . . . . . . . . 2795.6.2 Probe of deconnement . . . . . . . . . . . 2825.6.3 The NA50 experiment at CERN . . . . . . 2835.6.4 Dileptons and photons . . . . . . . . . . . . 286

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . 286

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CONTENTS xi

6 N = 1 Supersymmetric Gauge Theories 291Debashis Ghoshal

6.1 Introduction & the supersymmetry algebra . . . . 2916.2 Representations of supersymmetry on states . . . . 3026.3 Superspace & superelds . . . . . . . . . . . . . . 3106.4 Chiral & vector superelds . . . . . . . . . . . . . 3156.5 More on vector superelds . . . . . . . . . . . . . . 3216.6 Wess-Zumino model, supersymmetry breaking . . . 3256.7 Lagrangians of supersymmetric gauge theories . . . 3336.8 Supersymmetric QCD classical theory . . . . . 3406.9 Quantum corrections & effective action I . . . . . . 3476.10 Quantum corrections & effective action II . . . . . 3586.11 Appendix: Fierz identities . . . . . . . . . . . . . . 365

7 Introduction to AdS/CFT Duality 371Dileep P. Jatkar

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 3717.2 The large N duality conjecture . . . . . . . . . . . 3777.3 N = 4 Super-Yang-Mills theory . . . . . . . . . . . 3927.4 Collective coordinates . . . . . . . . . . . . . . . . 3987.5 Kaluza-Klein reduction . . . . . . . . . . . . . . . . 401

8 New Scenarios for Physics Beyond the StandardModel 413Sandip P. Trivedi

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4138.2 The ADD scenario . . . . . . . . . . . . . . . . . . 415

8.2.1 Gravity in extra dimensions . . . . . . . . . 4168.2.2 The size of extra dimensions . . . . . . . . 4188.2.3 The standard model lives on a domain wall 4198.2.4 The extra particles in the ADD scenario . . 4198.2.5 Kaluza-Klein harmonics . . . . . . . . . . . 420

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8.2.6 The Kaluza-Klein modes of the graviton andthe Newtonian potential . . . . . . . . . . . 422

8.2.7 Concluding comments . . . . . . . . . . . . 424

8.3 The Randall-Sundrum scenario . . . . . . . . . . . 4268.3.1 Spacetime in Randall-Sundrum scenario . . 4278.3.2 The energy scales in the RS model and the

hierarchy problem . . . . . . . . . . . . . . 4328.3.3 The Goldberger-Wise variation . . . . . . . 4378.3.4 Concluding remarks and outlook . . . . . . 443

9 Brane-World Phenomenology 449Sreerup Raychaudhuri

9.1 Gravity coupling to matter . . . . . . . . . . . . . 4509.2 Linearized gravity . . . . . . . . . . . . . . . . . . 4519.3 Gravitational elds in the ADD model . . . . . . . 4569.4 ADD Feynman rules . . . . . . . . . . . . . . . . . 4659.5 ADD phenomenology . . . . . . . . . . . . . . . . . 471

Contributors 483

Index 485

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Preface

The SERC Schools in Theoretical High Energy Physics, spon-sored by the Science & Engineering Research Council under theauspices of the Department of Science & Technology , Govern-ment of India have been held every year since 1985. Theseschools have served an extremely useful purpose in training stu-dents working for their Ph.Ds in various institutes and universi-ties across the country. Current research interests have been the

principal guide dictating the choice of courses, which have var-ied over the entire range from formal theory to phenomenol-ogy. The courses are designed so as to teach the basics ina thorough and systematic way, but at the same time give aavour and convey the excitement of the latest developments.More information is available on the homepage of the schools at:http://www.mri.ernet.in/ sercthep .

Lecture notes of previous SERC schools have only sporadi-cally been published. However, since we do recognize the valueof having the lectures available in print, which also makes themaccessible to a larger audience, we have collected, in this volume,the lectures from the XV and XVI SERC Main Schools in Theo-retical High Energy Physics held at the Saha Institute of NuclearPhysics, Kolkata and Harish-Chandra Research Institute, Alla-habad in 2000 and 2001 respectively. Although it has been overthree years since, the topics covered in these schools remain timely:this years Nobel prize in Physics recognizes the development of Quantum Chromodynamics. Regrettably, though, one set of lec-tures given at the HRI school could not eventually be included inthe present collection.

A number of people have helped, directly and indirectly, in the

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xiv Preface

preparation of this volume. Among them I would particularly liketo thank Rohini Godbole, Dileep Jatkar, Sunil Mukhi, SharmisthaMukhopadhyay, Sumathi Rao, V. Ravindran and all the contrib-

utors to this volume. It is a pleasure to acknowledge the helpextended by my colleagues at the Harish-Chandra Research Insti-tute in the organization of the XVI SERC Main School in THEP.Finally, I would like to thank Rohini Godbole, H. S. Mani andRam Ramaswamy, the editors of TRiPS, and HBA for patientlyputting up with several failed deadlines.

Debashis GhoshalHarish-Chandra Research Institute

Allahabad

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Chapter 1

Physics of Massive Neutrinos

Anjan S. Joshipura

1.1 Introduction

1.1.1 History

The existence of an almost massless neutral particle (later oncalled neutrino by Fermi) was postulated by Pauli in 1932 to ac-count for the continuous energy spectrum of the electrons emittedin nuclear decay. This particle was required to be a fermionin order to conserve angular momentum. Fermi incorporated thisparticle into a detailed theory of nuclear beta decay which couldaccount for the observed shape of the electron energy distribution

found in many nuclear beta decays. With availability of moreexperimental results, the original Fermi theory underwent manychanges and nally culminated into a simple and elegant V Atheory [1, 2] which universally describes all the known (charged)weak interaction processes at low energy [3, 4, 5, 6]. The V Atheory is basically an effective theory which allows reliable calcu-lations of weak interaction processes at energies O(100) GeV.The basic structure of this theory was later on generalized intoa full edged quantum theory based on ideas of spontaneouslybroken local gauge invariance [7]. It became possible to unifythe weak and electromagnetic interactions within this framework.

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2 1. Physics of Massive Neutrinos

The resulting theory is now known as the standard electroweakmodel. The neutrinos have played a very important role in ulti-mate formulation of this standard model (SM).

On the experimental side, neutrino was detected [8] in 1956,twenty-four years after its postulated existence. The detectionbecame possible due to the availability of intense neutrino beamfrom reactors and due to advances in electronics which made pos-sible an unambiguous detection of neutrino. Subsequent progressin neutrino physics was made after the development of intensebeams of muon neutrinos 1 at Brookhaven Laboratory. These neu-trinos are produced in the decays of high energy pions and kaons.It was found that neutrinos produced in these decays gave rise tomuons rather than to electrons showing that neutrinos producedin meson decays were different from the ones produced in betadecay.

After the discovery of b quark as a part of an SU (2)-doublet,and discovery of lepton, the existence of the third type of neutrino, , became imperative for consistency of the standardmodel. The lack of intense beam did not allow its directdetection for many years. It was nally discovered in 2001 atFermilab[9].

Neutrino physics has progressed a great deal in last ten years.The nuclear reactions taking place deep inside the Sun produceintense beam of neutrinos with a ux of 10 10 / cm2/ sec. It has be-come possible to precisely determine this ux over a wide energyrange

0.215 MeV. These experiments have helped in verica-tion of the standard solar model and have also given signicantinformation on neutrino masses and mixings. Intense beam of neu-trinos is also produced in our atmosphere by the incoming cosmicrays. These neutrinos have also been detected. It is commonlybelieved that detection of these neutrinos leads to a conclusionthat at least one neutrino is massive with a mass around 0 .1eV.In addition to the solar physics, neutrinos also inuence supernovadynamics, the dynamics of expanding universe and also provide

1 At many places we shall use the conventional terminology of collectivelydenoting neutrino and its antiparticle by the common name neutrino.

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1.1. Introduction 3

at least a part of the dark matter in the universe. Conversely, onecan use astrophysics and cosmology to learn about properties of neutrinos. Neutrino mass plays a very important role in all these.

These lectures notes are intended as a short introduction tothe subject of neutrino masses. We discuss experimental evidenceshowing that neutrinos are massive and also theories which canlead to the masses required on experimental grounds. Consideringthe available time limit, our main emphasis is on basic conceptsinvolved rather than the details. These details can be found inexcellent books [3, 5, 6, 10, 11] and review articles [12, 13, 14, 15].We proceed according to the following plan. This section con-tains a very brief historical introduction to weak interactions andthe standard electroweak model. The basic formalism of neutrinomasses is discussed in the next section. Here we discuss two com-

ponent neutrino hypothesis. Then we discuss the concept of theMajorana and Dirac neutrino masses. The nature of physical neu-trinos based on these mass terms is discussed in detail. Sectionthree and four present evidence for neutrino masses. The directmass determination (Kurie plot and neutrinoless double beta de-cay) is discussed in Sec. 3 and phenomena of neutrino oscillationsand related experiments are discussed in Sec. 4. This containsdiscussion of the solar and atmospheric neutrino anomalies. InSec. 5, we summarize the possible patterns of masses and mix-ing required experimentally. The nal section Sec. 6 contains abrief introduction on how can one theoretically obtain such mass

patterns.I thank the SERC school organizers for giving me an oppor-

tunity to give these lectures and Palash Pal for all the arrange-ments. I am grateful to Nimai Singh for helpful suggestions onthis manuscript.

1.1.2 Fermi theory and beyond

Fermi built a theoretical framework to describe nuclear beta de-cays. This was based on ideas in quantum electrodynamics. Allthe electromagnetic processes involving fermions result from the

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following basic interaction Lagrangian in quantum eld theory:

Lint = ie(x) (x). (1.1)Here ( x) describes the basic eld of a charged fermion ( e.g. , elec-tron). By convention ( ) creates an antiparticle (particle) anddestroys particle (antiparticle). A(x) describes electromagneticeld which creates and destroys photons. The above interactionleads to processes in which a pair of fermions are attached to aspacetime point and photon (or fermions) connect different spacetime points. The beta decay involves four fermion elds. Fermidispensed with the photon eld and assumed that all the fermionsin beta decay are created or destroyed at a single space time point.This is described by the following four fermion Lagrangian:

LF = GF 2 e(x) (x) p(x) n (x), (1.2)

where x refers to quantum eld corresponding to x=electron ( e),proton ( p), neutron ( n) and neutrino ( ) respectively and GF / 2is a constant with the dimension 2 of inverse (mass) 2. Its value isapproximately given by 1 .16 105 GeV2. The above interactioncan be used to calculate rates and shape of beta decay spectrumand was found to be quite successful in describing a large class of nuclear beta decays. But it could not describe all the beta decays.In the non-relativistic limit (appropriate here since nucleon energy

is much smaller than its mass), Eq.(1.2) does not contain nuclearspin operator. As a result, it cannot describe nuclear transitions(called Gamow-Teller transitions) which involve change in nucleonspin. This is accomplished by means of the following interaction:

LGT =GF 2 e(x) 5 (x) p(x)

5n (x). (1.3)

Eqs.(1.2,1.3) are special cases of the following most gen-eral four fermion interaction which could be written down using

2 We follow the conventional notations and set h = c = 1.

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1.1. Introduction 5

Lorentz invariance:

Lgen =GF 2

i

e(x) i (C i + 5C i ) (x) p(x)i n (x), (1.4)

where the sum over i refers to sum over linearly independent com-binations of gamma matrices namely, i = 1 , , , 5, 5 re-ferred to respectively as scalar (S), vector (V), tensor (T), axialvector (A) and pseudo-scalar (P). Parity violation found experi-mentally is already contained in the above interaction. A priori,the above interactions contain 10 independent parameters and itmay seem difficult to determine these parameters experimentally.In the early days, it was found that the nuclear beta decays werefavouring the S, T and P interactions while the pion and muondecays were clearly showing V

A structure. Very careful analysis

[1, 2] of the available information led to the proposition that thesignatures of the S, T and P interactions may not be well-foundedexperimentally and all the available interactions may actually beV A. This proposition was theoretically derived from chiralinvariance, i.e. , invariance of Eq.(1.4) under independent chiraltransformations 5 on each of the elds. This invarianceamounts to saying that particles ( e.g. , electrons, neutrinos) emit-ted in a decay are left-handed and their antiparticles are right-handed. This was experimentally veried for the electron andalso for the neutrino by direct helicity measurements[3]. It can beshown that the chiral invariance reduces the above 10 parameter

four fermion interaction to only one parameter V A structuregiven by

LV A =GF 2 i e(x) (1 5) (x) p(x)

(1 5)n (x),(1.5)

As the evidence gathered, the validity of V A interaction wasproven to be more general. It was found to describe leptonicweak interactions ( e.g. , and decay), semi-leptonic weak inter-actions ( e.g. , and K decays to leptons) and non-leptonic decaysof baryons and mesons ( e.g. , K decays to pions). The SM treats

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quarks as fundamental and thus gives a Lagrangian which con-tains quark elds instead of the nucleon elds as in Eq.(1.5). Thecomplete four fermion interactions which generalizes Eq.(1.5) is

given byLV A =

GF 2 J + J , (1.6)

where

J = e,, (1 5)l +

i,j =1 ,2,3(U CK M )ij u i (1 5)d j ,

(1.7)with l denotes three charged leptonic elds and U CK M is a 33unitary matrix and i, j run over three generations. A remarkablefeature of the above equation is its universality. All the weakinteractions among leptons are given in terms of a single parameterGF / 2. The same parameter also determines weak interactionsof quarks apart from the elements of mixing matrix U CK M whichgeneralizes the original Cabibbo current proposed to describe thestrangeness changing weak interactions. Such universality cannotbe accidental. It is now known that this is closely linked to thelocal gauge invariance under an SU (2) group of transformations.This requires the existence of a charged spin 1 bosons W withthe following interaction:

LW =g

2 2(J + W + h.c.). (1.8)

Here g is the fundamental gauge coupling analogous to the electro-magnetic charge e. The above interaction leads in the low energylimit E

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1.2. Neutrino masses: basic formalism 7

W 3 corresponding to the third component of the angular momen-tum. The current J 3 coupled to W 3 is given by the commutatorof J . It is seen from Eq.(1.7), that J 3 determined this way doesnot coincide with the electromagnetic current. In order to in-corporate electromagnetism one needs an additional U (1) groupwith corresponding gauge boson B and its gauge coupling g . Thephysical photon eld is a combination of B, W 3 dened as

A = cos W B + sin W W 3 ,

where angle W is determined to be tan W = g /g by requiringthat the current coupled to A be the electromagnetic current.The orthogonal combination of B and W 3 is conventionally de-noted by Z and it leads to additional weak interactions. Struc-ture of this neutral weak interactions is determined by the algebraof the SU (2) U (1) group and is given as follows:

LZ =g

2cos W f f [(T 3f 2Qf sin2 W ) T 3f 5]f Z . (1.9)

Eqs.(1.8) and (1.9) describe complete weak interactions of allfermions in the standard model. Thus various neutrino produc-tion mechanisms which we will be talking about are determinedby these equations. The new ingredient which is not part of theseequations and of the Standard Model is neutrino mass. We turnto this in the next section.

1.2 Neutrino masses: basic formalism

As we mentioned, neutrinos produced in any experiments arefound to be left-handed while antineutrinos are right handed par-ticles. This fact led to a two component hypothesis for neutri-nos. According to this hypothesis, neutrinos are described interms of elds having two (complex) components unlike other spin1/2 fermions which need four component elds. Two componentdescription of neutrino emerges naturally from the conventionalDirac equation in the massless limit. We discuss this rst and use

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this picture to build up formalism to describe neutrino masses.This formalism is used to describe three basic types of neutrinomasses: Majorana, Dirac and pseudo-Dirac.

The evolution of a free massive neutrino is described by theDirac equation(i m) = 0 . (1.10)

The gamma matrices ( = 0 , 1, 2, 3) are chosen to satisfy

= 0 0, 5 = i 0 1 2 3 = 5 = 5 , (1.11)

in addition to the usual anti-commutation relations. We shalladopt throughout the following specic representation called thechiral representation for the gamma matrices:

0 = 0 11 0

, = 0

0, 5 = 1 00 1

.

(1.12) represent here the conventional 2 2 Pauli matrices. The 5is diagonal in this representation and is used to dene two chiralcomponents:

f L =12

(1 5) f, f R =12

(1 + 5) f, (1.13)

for any fermion f . The Dirac equation can be converted to thefollowing equations in the massless limit:

i L,R = 0 ,

.p1 5

2 =

1 52

, (1.14)

where p denotes unit vector along the momentum direction and.p denotes component of spin along the direction of motion.The second of the above equations is known as the Weyl equa-tion. These equations show two important properties of masslessfermions. First, the dynamics ( i.e. , time development) does notmix L and R components. Thus these two components can be

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regarded as describing two independent particles with no connec-tion between them. Secondly, L ( R ) carry negative (positive)helicity. As we shall show, if neutrino is left-handed then the

anti-neutrino ( i.e. , corresponding charge conjugate eld) is righthanded. Both these particles can be describe by a single eld Lwhich in eld theory, represents creation of antineutrino and de-struction of a neutrino. Thus massless neutrinos are completelydescribed in terms of L alone and one does not need the eld R . This was the basic idea behind two-component picture of neutrino which was originally proposed to describe massless neu-trinos in parity violating theory. But even massive neutrinos canbe described in terms of two component elds. To understand thismore clearly we now discuss various types of neutrino mass terms.

1.2.1 Neutrino mass termsNeutrino mass corresponds to a Lorentz invariant renormalizableterm in the Lagrangian connecting a left and a right-handed eld.There exists possibilities of writing two independent mass termsin case of a neutral fermion. These two different possibilities aretermed as Dirac and Majorana masses. In order to introduce thesemass terms, we need to discuss the charge conjugation propertyof the neutrino eld.

Charge conjugation symmetry relates particles and antiparticles.A four component neutrino eld transforms under charge conjugation as follows:

c C C 1 = C T . (1.15)The Lagrangian for free neutrino eld remains invariant if thematrix C is chosen to satisfy

C C 1 = T .C can be chosen to be i 2 0 in the specic representation (1.12)for the gamma matrices. In this case one has

c = i 2 0 T = i 2 . (1.16)

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The matrix C satises

C = C 1 = C T = C. (1.17)The following relations are easy to prove using properties of

gamma matrices:

cL P L c = C T R cR , cR P R c = C T L cL . (1.18)

It follows that charge conjugate cL of a left-handed eld L is aright handed object and vice versa. Thus if neutrino emitted ina beta decay is left handed then the corresponding antineutrinowould be right handed. The important point to keep in mind isthat although the cL is right handed, it does not coincide with Rwhich as mentioned earlier is an independent eld with its owndynamical evolution.

A left handed neutrino eld L can form a mass term eitherwith its charge conjugate (and hence right handed) eld cL or itcan combine with an independent eld R . Moreover, R can alsocombine with its left-handed charge conjugate cR to give a massterm.

Let us consider a theory containing two independent elds Land R . Here R is an independent eld and thus is not the chargeconjugate of L . The latter would generally represent any of theneutrino elds corresponding to active ( i.e. , those having weakinteractions) neutrinos e,, . R can represent a right handed eld

unrelated to any of these. Such R would transform as a singletunder SU (2) U (1). Alternatively, R can be charge conjugateof any of the active neutrinos, e.g. , L may represent eL and Rmay be cL . We allow both these possibilities.

We can write the following mass terms between L and R :

Lmass = L mD R +12

mL L cL +

12

mR cR R + h .c., (1.19)

where we have used primed elds to distinguish them from themass eigenstates to be introduced soon. The terms with coeffi-cients mL,R are known as the Majorana mass term and the mD

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term is known as the Dirac mass term. The Majorana mass termsare not invariant under global phase changes of the elds since L,R and

cL,R change by the same amount under this transfor-

mation. In contrast, the Dirac mass term can be made invariantif L and R are transformed by the same phase. Thus Majo-rana mass terms violate lepton number while the Dirac mass termrespects it.

It is seen from Eq.(1.19) that neither L nor R is a masseigenstate. The nature of physical neutrino is determined by goingto the mass basis. To do this, we rewrite Eq.(1.19) as follows:

Lmass =12

L cR

mL mDmD mR

cL R

+ h.c. (1.20)

We have made use of the following relation in writing the aboveequation.

L R = cR

cL . (1.21)

This relation is a special case of a more general identity which isquite useful in many of the algebra related to charge conjugateelds:

i = cC T i C 1c, (1.22)

where , are any two Dirac spinors, i represents products of the Dirac gamma matrices. Let us rewrite Eq.(1.20) as:

Lmass =12 n LM n

cL + h.c., (1.23)

where nL ( L , cR )T denotes a column vector for two neutrinostates and M is a 22 matrix dened as:

M mL mDmD mR

. (1.24)

It is possible to diagonalize M through a unitary matrix U U T M U = diag ( m1, m 2), (1.25)

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where m1,2 are eigenvalues of M given bym1,2 =

12

mL + mR

(mL mR )2 + 4 m2D . (1.26)

Note that m1,2 dened above are not necessarily positive. Thephysical neutrino mass basis are dened as

L cR U

1L c2R

. (1.27)

The new states 1L and 2R represent chiral components of twodifferent neutrino states with masses m1 and m2 respectively. If CP conservation is assumed, U can be taken as an orthogonalmatrix which is specied in terms of a mixing angle giving us

1L = cos L sin cR , 2R = sin cL + cos R , (1.28)with

tan2 =2mD

mR mL. (1.29)

Since masses m1,2 can have either sign, let us dene m i |m i |i(i = ) and rewrite Eq. (1.23) as:

Lm =12

(|m1|1 1L c1L + |m2|2 c2R 2R + h.c. ) , (1.30)

where we made use of Eq.(1.25).We have been writing all mass terms in terms of chiral projec-tions of the eld. We can always dene appropriate four compo-nent objects and write masses using these new elds. Dene

1 = 1L + 1 c1L , 2 = 2R + 2 c2R . (1.31)

Eq.(1.30) assumes the following form:

Lm =12

(|m1| 1 1 + |m2| 2 2) . (1.32)

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From the denition Eq.(1.15), of the charge conjugation, it is ob-vious that the elds 1,2 satisfy

c

1,2 = 1,2 1,2.Thus both the elds 1,2 are self-conjugate. Neutrinos describedby these elds are called Majorana neutrinos.

We started with two independent two component objects L , R with the most general mass term given by Eq.(1.19). Thistheory could be rewritten in terms of two (four component) objectssatisfying Majorana condition. Eq.(1.19) therefore generically de-nes Majorana neutrinos. However there are special cases whichare of considerable theoretical importance. We discuss these casesnow.

1.2.2 Dirac neutrinoThis case corresponds to the limit mL = mR = 0. The eigenvaluesm1,2 in Eq.(1.26) are equal and opposite thus 1 = 2 . We canthen dene

= 1 + 2 2 .

The mass term can then be rewritten as

Lmass = |m1|. (1.33)By denition, = c and the above mass term describes a four

component Dirac fermion. The above mass term is invariant undera phase transformation on . This phase transformation may beidentied with the lepton number which is conserved in the limitmL = mR = 0. In this limit the two original Majorana elds 1,2have merged into a Dirac state .

1.2.3 Pseudo-Dirac neutrino

Two eigenvalues turn out to be equal and opposite also in thespecial case mL + mR = 0 but mL,R = 0. Due to equality inthe masses, the original mass term can be converted to a Dirac

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mass term by dening exactly as above. But there is a subtledifference between these two situation. In both cases, the nalmass term as given in Eq.(1.33) is invariant under lepton num-

ber symmetry. But if we look back at the original mass matrixin Eq.(1.24) then we nd that this matrix is invariant under lep-ton number symmetry if mL = mR = 0 but is not invariant if only mL + mR = 0 3 . Due to non-invariance of the mass matrixlepton number is violated [16] by the full Lagrangian although itis respected by Eq.(1.33). As a result, the two different compo-nents 1 and 2 of receive different radiative corrections andthe Dirac neutrino gets split. Thus the nal theory contains apair of Majorana neutrinos with almost degenerate masses. Sucha pair is referred to as pseudo-Dirac neutrino. Phenomenologicalimportance of this case is discussed in [16, 17].

1.2.4 Seesaw masses

Now let us consider the limit mL = 0 , m R mD . The mR is typi-cally assumed to be much larger than the electroweak scale. Sincethis mass corresponds to the Majorana mass for the right-handedeld R , this eld has to be identied with some SU (2) U (1)singlet eld unlike in the previous two cases, where R may repre-sent charge conjugate of the non-singlet neutrinos. The neutrinomasses are given in this case by

m1

m2DmR

; m2

mR

.

Thus neutrino masses are hierarchical in this case. This schemeis ideal and one of the most preferred schemes for description of neutrino masses. In this case, very small mass for neutrino is gen-erated through a high scale mR . The high scale can be naturallyintroduced in grand unied theory such as SO (10). Moreover,

3 In fact, it is not possible to dene any U (1) symmetry under which theneutrino mass matrix remains invariant in this case. The occurrence of Diracneutrino at tree level in this case is a consequence of the invariance of Eq.(1.24)under a discrete symmetry, see [16] for a detailed discussion on this point.

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1.3. Neutrino mass measurements: direct detection 15

the angle is very small and the light neutrino is predominantly L and heavier is mainly R , see Eq.(1.28). Thus this schemeautomatically explains why the conventional neutrinos are much

lighter than other fermions. We will talk more about this schemein our subsequent discussion.

1.3 Neutrino mass measurements: direct detection

We will briey discuss various experimental methods employed todetect neutrino masses. We shall omit many of the details whichcan be found, for example, in [5, 6]. The experiments which lookfor neutrino masses can be divided in two categories. Some exper-iments are capable of obtaining information on the absolute neu-trino masses. This is done through kinematical studies of heavyparticle decays which produce a neutrino in the nal state. Thenuclear beta decay and neutrinoless double beta decays provideinformation on the electron neutrino mass in this way. The massesof and can be measured through the pion and tau decays.Direct information on neutrino masses can also be obtained in theneutrino signal from supernova. All these experiments have so fargiven only upper limits on the neutrino masses.

The second category of experiments study neutrino oscilla-tions and probe neutrino (mass) 2 differences. Three sets of ex-periments in this category have given us positive information onthe neutrino (mass) 2 differences. These correspond to the solar

and atmospheric neutrino experiments and the laboratory exper-iments at Los Alamos. We shall briey talk about them in thissection.

Primary requirement of oscillation experiments is an intenseneutrino beam. These beams are either produced in laboratory orarise from some natural sources. We have the following possibili-ties:

Nuclear reactors produce intense beam of the electron anti-neutrinos. The average energy of these neutrinos is aroundfew MeV.

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One can produce intense beam of the muon neutrinos andantineutrinos in laboratory experiments. Very energetic pro-tons hitting a target produce large number of pions and

kaons. They mainly decay to muons and their neutrinosgiving an intense beam of the muon neutrinos. The muonsproduced this way also decay and lead to large number of electron neutrinos.

The fusion reactions inside the Sun gives rise to electronneutrinos. These neutrinos carry energies of O(110) MeV. Reactions similar to the ones in item (2) also occur in naturewhen energetic cosmic rays enter our atmosphere. This gives

natural beam of muon and electron neutrinos.

Intense beams of all types of neutrinos come from supernovaexplosion. Detection of this however requires explosion tooccur not far away from our galaxy and so far there has beenonly one explosion which has given us detectable neutrinosignal.

The neutrino beams obtained through above methods havebeen used to study neutrino oscillations in laboratory. The oscil-lating neutrinos are detected through their charged and neutralcurrent interactions. Different experiments use different methodsfor their detection and we shall describe them as we go along.

1.3.1 Beta decay and m eThe shape of the electron energy distribution in the nuclear betadecay

N (A, Z ) N (A, Z + 1) + e + e (1.34)is sensitive to the neutrino mass. The shape function is conven-tionally dened as

K (E ) d/dE

pE F (Z, E )1/ 2

(Q + m e E ) (Q + me E )2 m2 1/ 2 ,(1.35)

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1.3. Neutrino mass measurements: direct detection 17

where p, E respectively denote the electron momentum and en-ergy. d/dE is the differential probability for the electron emis-sion. Function F (Z, E ) arises because of the coulomb interactionsof the departing electron and Q denotes the Q value of the betadecay. It is seen from the above equation that the plot of K (E )versus E (known as Kurie plot) is a straight line in the absenceof neutrino mass. The neutrino mass makes this curve bend nearE Q, the departure being more prominent for lower values of Q. Thus decays with lower Q can give better limit on m . Thebeta decay of Tritium is used to obtain limit on m due to its lowQ value (18.6 KeV). The best limit obtained in this decay is givenby[18]:

m 2.2 eV. (1.36)The quoted limit is obtained by assuming very small mixing of

the electron neutrino with other neutrinos.

1.3.2 Neutrinoless double beta decay

It can happen that ordering of the nuclear energy levels in somenuclei does not allow the ordinary beta decay to take place. Forexample, the ground state of 7632Ge lies lower than that of 7633Asto which it could have transformed by ordinary beta decay. Theformer is however higher than the ground state of 7634Se . Thus7632Ge can decay to 7634Se by emitting two electrons. Such type of processes

N (A, Z ) N (A, Z + 2) + 2 e + 2 e (1.37)are called double beta decay. These processes are second order inFermi coupling and are very rare but they have been observed.

It turns out that the two neutrinos emitted in the above pro-cess can be made virtual if they are massive. This leads to thefollowing process known as neutrinoless double beta decay

N (A, Z ) N (A, Z + 2) + 2 e (1.38)Two features of the above process distinguish it from Eq.(1.37).The lepton number is violated here by two units unlike in Eq.(1.37)

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which conserves lepton number. Secondly, sum of the energies of two electrons is xed by kinematics. Thus one nds a peak inthis energy distribution. This is a distinguishing feature of the

neutrinoless double beta decay which is used as a signal for thisprocess.The lepton number violating neutrino mass probed in this pro-

cess is given by

m ee =i

U 2ei m i . (1.39)

Neutrino masses are denoted here by m i with i = (1 , , 2, 3). U isneutrino mixing matrix analogous to U CK M dened in Eq.(1.7).

The presence of a non-zero mee has been searched for innumber of experiments. The most stringent limit obtained so farcomes from the Heidelberg-Moscow collaboration [18]:

m ee 0.38eV , (1.40)This limit is stronger than the limit on absolute neutrino massobtained from the Kurie plot in Eq.(1.36). But both limits areconsistent since there can be cancellations between individual neu-trino mass in Eq.(1.39).

Eq.(1.39) not only measures the effective neutrino mass butalso provides information on the nature of neutrino. Let us con-sider two neutrino mixing as in previous section. The effectivemass is given in this case by

mee = m1 cos2 + m2 sin2 . (1.41)

If both neutrinos are Majorana particles with unequal masses thenm ee is non-zero and measures the weighted mass as given by the

RHS of above equation. When two masses are equal and oppositeand mixing angle is / 4, two neutrinos can be regarded together asa Dirac neutrino with exact lepton number conservation. m ee inEq.(1.41) vanishes in this case as expected. If neutrino is pseudo-Dirac then m1 = m2 at tree level but is not / 4. m ee is againnon-zero in this case since there is no lepton number conservationin this case. The m ee can be used together with other evidence

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1.4. Neutrino mass measurements: oscillations 19

for neutrino masses to learn about the nature of neutrino spectrumand there has been extensive studies of this topic [19].

1.4 Neutrino mass measurements: oscillations

Search for neutrino oscillations has proved to be the most powerfulway of obtaining information on neutrino masses. Neutrinos areproduced through their charged weak interactions given :

LW =g

2 2 l L L W + h.c. . (1.42)

= e,, runs over the physical avour states correspondingto the mass eigenstates of the charged lepton. The denedabove do not posses denite mass but is given by a combination

of neutrinos i having mass m i .

= U i i (1.43)

Neutrinos produced through their charged current interactionsare therefore superposition of neutrinos with denite masses eachmass states evolving with different phase factor corresponding toits energy. This leads to the phenomena of neutrino oscillations.Let us consider two neutrinos e, which would be produced alongwith the electrons and the muons respectively. These are given interms of the mass eigenstates by:

e = cos 1 + sin 2 , = sin 1 + cos 2 . (1.44)

e produced at t = 0 evolves to

| e(t) = cos eiE 1 t | 1 + sin eiE 2 t | 2at time t. The neutrino energies are given for the relativisticneutrinos by

E i p +m2i p

.

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As long as the neutrino masses m i are not equal, relative strengthsof 1 and 2 in original e beam vary with time and the e statedoes not remain orthogonal to the dened by Eq.(1.44). As a

result, a non-zero amplitude to nd a in the original beam of e develops after a time t (or distance L). The probability P e tond = e , in the original e beam is given by

P ee = 1 sin2 2 sin2 m212L

4E ,

P e = sin 2 2 sin2 m212L

4E , (1.45)

where m212 m22 m21. It is clear that neutrino oscillation prob-abilities measure their (mass) 2 differences and mixing angle, bothof which are required to be non-zero. The conversion probabilityP e is signicant if sin 2 2 is large or

m 212 L4E / 2 or both. Inthis case P e can be used to obtain information on m212 and/or

sin2 2. One can increase the sensitivity of m212 measurement if Lis large and/or E is small. The neutrino intensity however drops as1/L 2. Thus intense beam, small energy and large path lengths arebasic criteria needed to probe smaller (mass) 2 differences throughneutrino oscillations. The solar neutrino beam proves to be idealfor this purpose and can measure neutrino (mass) 2 differences assmall as 1010 eV2. We summarize sensitivity of various experi-ments in Table 1.1 given below.

As seen from Table 1.1, different experiments are sensitive todifferent values of neutrino masses. The original beam employedis also different either electron or muon type. Thus these experi-ments search for oscillations of different neutrinos. These searchesare made in two ways. One direct way is to start with a neutrinoof given avour and look for the appearance of a different avourby detecting its charged lepton. This method may not always befeasible. For example, the reactors give electron neutrinos buttheir energy is not sufficient to produce muons. Thus even if re-actors e have oscillated to , we cannot easily detect these .In such cases, one looks for reduction in uxes of the original neu-

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Type of Typical Typical m2

experiment energy (MeV) path length probed (eV 2)10 m-1 km

Reactor few MeV 100 km 103

1for KamLandAccelerator 10 3 104 100-1000m 1102

Solar 0.110 108 km 1010 1011Atmospheric GeV 10

4 106m 102 103

Table 1.1: Neutrino oscillation experiments and their sensitivity tomeasurement of the difference in mass-squared between a neutrinopair.

trino avour which is over and above the normal 1 /L2

reduction.This type of experiments are known as disappearance experiments.Both these types of experiments have helped us in nding valuesand in large number of cases restrictions on neutrino mixing an-gles and (mass) 2 differences. We briey summarize below positiveinformation that has come from neutrino oscillation experiments.

1.4.1 Atmospheric neutrinos

Cosmic ray interactions with our atmosphere produce very ener-getic pions and kaons. These particles decay mainly to muons and

their neutrinos. Most muons produced in these decays also decayto one electron neutrino and one muon neutrino. Thus one hasroughly two muon neutrinos and one electron neutrino for everypion or kaon that decays. Flux of these neutrinos is reasonablylarge, roughly 100 atmospheric neutrinos pass through our bodyevery second. But due to their weak interactions, only one suchneutrino can interact with our body in thousand years! It requirestherefore very large mass to detect atmospheric neutrinos.

The detection of atmospheric neutrinos was made in a mas-sive detector called Super Kamiokande[20] in Japan which con-tains 50,000 tons of pure water. This detector not only detects

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these neutrinos but can also distinguish between the electron andmuon types of neutrinos and thus can detect oscillation of the at-mospheric neutrinos. Typical energies of atmospheric neutrinos

is 1 GeV or more. Thus electron and muon type neutrinos giverise respectively to energetic electrons and muons after interact-ing with water. These charged particles travel faster than speedof light in water and give out Cerenkov radiation. These light isdetected by photomultiplier tubes surrounding the detector. Theneutrino energy in a given event is inferred from the amount of light collected in photomultiplier tubes. The cone generated bythe Cerenkov light of muons and electrons produce different typesof rings in the detector which is used to differentiate between theelectron and muon neutrinos. The charged particles travel almostin the same direction as neutrinos. One therefore has informationon avour, energy and direction of neutrinos which has been usedto conclude the presence of neutrino oscillations.

The observations at Super Kamioka detector showed that the are oscillating while e do not appear to be oscillating withsimilar wavelength. These observation suggest that are oscil-lating to . It is possible to detect neutrino oscillation in thisexperiment since different neutrinos travel different distances be-fore entering the detector. Those coming from top travel only inthe atmosphere and those coming from below also travel insidethe earth -the exact distance traveled is a function of the zenithangle. The observed ux at Superkamioka showed variation withthe zenith angle which is in accord with the oscillation probability:

P = 1 sin2 2 sin2 m223L

4E , (1.46)

where m223 m23 m22 is the (mass) 2 difference between themu and the tau neutrinos. The maximum path length corre-spond to the earth diameter. This distance when translatedthrough Eq.(1.46) for 1 GeV neutrinos correspond to m223 of about 10 3 eV2. The mixing angle between the mu and tauneutrinos is found to be nearly maximal.

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Reaction E max Flux (10 10 /cm 2sec) p + p 2 H + e+ + e 0.42 5.947Be + e 7 Li + e 0.86 (90%) 0.48

0.36 (10%)8B 8 Be+ e+ + e 14.06 5.5 104

Table 1.2: Important nuclear reactions responsible for the produc-tion of the solar neutrinos. The ux of neutrinos produced in eachreactions and their highest energies are also shown in the table.

1.4.2 Solar neutrinos

The surface of our atmosphere receives = 1 .4 106 erg of en-ergy per cm 2 every second. This energy is the result of thermonu-clear reactions occurring inside the Sun which converts 4 hydrogenatoms into a Helium atom through complicated chain reactions.Each such conversion is accompanied by 26 MeV of energy andtwo neutrinos. Thus one neutrino is emitted for every 13 MeV of thermal energy received. The ux of neutrinos coming from theSun is then given by

e

13MeV 6 1010 e / cm2/ sec.This is quite intense source of the electron neutrinos which never-theless requires very large underground detectors to detect them.

By now, these neutrinos are detected in four different types of experiments (1) Cl experiment at Homestake [21] (2) Gallium ex-periments Gallex [22] and SAGE [23] (3) Elastic scattering exper-iments Kamioka and Superkamioka [20] and (4) Sudbury detectorSNO [24]. We briey describe results of these experiments andthen turn to consequences of experimental results. About 98% of total neutrinos are produced in the reactions shown in Table 1.2.

The neutrino spectrum ( i.e. , variation of ux with energy) forsolar neutrino in these reactions is determined by the standardnuclear physics. The normalization of each of these uxes is de-termined in the standard solar model which takes into account

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Experiment Reaction Threshold exptSS M (in MeV)

SAGE,GALLEX, e + 71 Ga 71 Ge + e 0.233 0.584GNO 0.039Homestake e + 37 Cl 37 Ar + e 0.814 0.335

0.029Superkamioka xe x e 5.5 0.459

0.017 xe x e 5.0SNO eD p + pe (CC) 0.347

0.029 x D n + n x (NC ) 1.00720.126

Table 1.3: Characteristics of different solar neutrino experimentsand the neutrino ux measured by them. The uxes are normal-ized with respect to the standard solar model.

dynamics and conditions existing in the Sun. Experiments men-tioned above have different thresholds and thus detect differenttypes of neutrinos. Salient aspects of these experiments are sum-marized in Table 1.3 below.

Some noteworthy points are:

Cl and Ga experiments do not have directional and timesensitivity. They only measure the overall conversion rates. The Gallium experiments are the only experiments measur-ing the most dominant pp neutrinos because of their low

threshold.

As shown in the last column, all the experiments nd sup-pression in the neutrino ux compared to the standard solarmodel. This suppression shows that a part of the solar neu-trinos are getting converted to some other neutrino avour,

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e.g. , muon or tau neutrino. This converted neutrinos cannotproduce the corresponding charged lepton because of theirlow energy and thus produce apparent decit in the solar

ux.

The amount of decit seen in different experiments is differ-ent. This shows that mechanism which suppresses the solarux is energy dependent.

The Cl and Ga experiments are sensitive only to thecharged current process. The elastic scattering rates in Su-perkamioka and SNO receive a sub-dominant contributionalso from the neutral current process ( + e + e) butit is not possible to separate these two contributions directly.

SNO experiment has separately measured the charged cur-rent process e + D p + p + e and the neutral currentprocess + D p + n + . This experiment thereforemeasures the electron neutrinos as well as the neutrinos towhich part of the solar neutrinos get converted. The sum of the charged current and neutral current uxes agrees withinerrors with the ux predicted in the standard solar model.

1.4.3 Mechanisms for solar neutrino conversion

The above empirical facts can be understood in terms of differentmechanisms. We discuss below two of the most popular possibili-

ties.

1.4.4 Vacuum oscillations

It is assumed in this mechanism that the solar neutrinos oscillateto other avour during their journey to earth from the point of production deep inside the core. The probability P ee that theelectron neutrinos retain its avour after traveling a distance Lis given by Eq.(1.45) with L 10

8 km. corresponding to theSun-Earth distance. In case of the MeV neutrinos, the energydependent factor in P ee averages out if m212 is 1010 eV2.

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This corresponds to energy independent survival probability. TheP ee depends on energy if m21210

10 1011 eV2.The detailed analysis of the experimental results assuming vac-

uum oscillation has been made over the years [25]. The latestglobal analysis using all results from different experiments [26]does not favour this possibility as an explanation for the decit inthe solar ux.

1.4.5 MSW mechanism

This mechanism also requires non-zero mixing and (mass) 2 differ-ence between two neutrinos as in vacuum oscillation but details of conversion to a different avour are very different. It is possibleto explain the solar decit in this mechanism for a different valueof (mass) 2 than in the previous case. Unlike in the vacuum case,the neutrino interactions with the solar matter plays an importantrole here.

As discussed before, the neutrino produced in some reactionat time t = 0 has non-zero amplitude to be found as e or at alater time t. Let us denote the neutrino state at time t by

| e(t) = ae(t)| e + a(t)| . (1.47)The evolution of a (t) is governed by the free particle Hamiltonian

which can be written as

iddt

aea

=1

2E m212 cos2 12 m212 sin212 m

212 sin2 0

aea

,

(1.48)where m212 m22 m21. Derivation of above equation is straight-forward and standard [6]. Although it looks different in form,the above equation can be shown to lead to the same oscillationprobability as given in Eq.(1.45) when neutrino travels throughvacuum. But solar neutrinos do not do that. They interact withsolar matter. The effect of these interactions is to modify evolu-

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tion equation to

id

dt

ae

a=

1

2E

A m212 cos2 12 m212 sin212 m

212 sin2 0

ae

a,

(1.49)where A = 2 2GF n eE denotes the effective (mass) 2 differenceacquired by neutrino pairs because of their interactions in theSun. ne which denotes the electron number density inside theSun is a function of the distance from the core inside the Sun.This makes A dependent on the distance traveled by the neutrinoinside the Sun. The value of A at the solar core is about 10 5 eV2.This additional (mass) 2 changes as neutrino travels inside the Sunand it also depends upon the energy of neutrinos. If m212 isof O(105) eV2 then it could happen that the matter-dependent(mass) 2 for some of the neutrinos matches exactly with m2

12at some point inside the Sun. When this happens, the diagonalterms in Eq.(1.49) become equal and effective neutrino mixingbecomes maximal even if it was very small to start with. Thisresonance enhancement of the mixing in the presence of matterwith varying density is termed as Mikheyev-Smirnov-Wolfenstein(MSW) effect.

The MSW effect can result in strong reduction in the neutrinoux. This can be seen in a simple situation as follows. Assumethat the vacuum (mass) 2 difference of neutrino pair is smallerthan the value of A at the core In this case, the electron neutrinoproduced at the core is largely the heavier mass state 2. Thisneutrino goes through resonance and emerges as a 2 state in theadiabatic approximation. But the probability to nd the electronneutrino in the state 2 at the detector in vacuum can be read off from Eq.(1.44) and is given by

P ee = sin 2 .

This shows that one could obtain sizable suppression in neu-trino ux through adiabatic transition in case of large mixing an-gle. This survival probability appears to be energy independent.But the above expression for P ee holds only for the neutrinos which

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undergo resonance transformation. Moreover, non-adiabacity inneutrino transition adds another energy dependent factor in P ee .As a result, the MSW effect is found to describe all the available

experimental results quite well. We refer the reader to a detaileddiscussion on P ee to reviews [25] and books [6, 5].

Quite extensive work has been done to test the MSW mecha-nism against experimental results. The latest analysis [26] impliesthat the most likely solution for the solar neutrino decit is theMSW solution with large mixing angle tan 2 0.2 0.8 and m2122 105 eV2 3 104 eV2 at 3 level.

Positive result in search of neutrino oscillations also comesfrom a detector called LSND [27] at the Los Alamos laboratory.This detector uses a beam and looks for its oscillations to e bytrying to detect positron and delayed neutron produced by the re-action e + p e+ + n. This experiment reported evidence for the e oscillations corresponding to a (mass) 2 difference of O( eV2)and mixing sin 2 2103. This value is quite different from thescales found in solar and atmospheric experiments. Most regionof parameter space allowed by the LSND experiment is alreadyruled out by other experiments, e.g. , KARMEN[28]. LSND evi-dence is therefore required to be checked through an independentexperiment.

Apart from these positive results, there also exists a strongnegative result which provides quite useful information on neu-trino mixing. This comes from the reactor experiment at CHOOZ.According to it, probability (1 P ee ) for the electron (anti) neu-trino to convert to any avour is required to be less than 0.04if the corresponding (mass) 2 difference is around the atmosphericscale4 .

We now discuss possible neutrino mass patterns implied bythese results.

4 In general there is an exclusion curve [29], which gives bound on 1 P eeas a function of (mass) 2 difference and mixing angle.

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1.5. Neutrino mass spectrum from experiments 29

1.5 Neutrino mass spectrum from experiments

We discussed experimental results in previous section assuming

mixing between two generations. We found that one needs threenon-zero and hierarchical (mass) 2 differences in order to accommo-date results from the solar, atmospheric and LSND experiments.Since three neutrino states can have only two independent (mass) 2

differences, all the experimental results cannot be reconciled withthe known neutrino avours and one would need a fourth lightneutrino state. Since the LSND results are not as rmly estab-lished as other two, we will discuss only three neutrino picturewhich can explain the solar and atmospheric decits.

We need neutrino masses to be such that one (mass) 2 differencecorresponds to the solar scale 10

5 eV2 and the other to theatmospheric scale atm . This implies one of the following threepossibilities:

(A) Hierarchical masses: m21 m22 m23 atm ;(B) Inverted hierarchy: m21m

22 atm m

23; m22m21;

(C) Almost degenerate masses: m21m22m

23 atm .

All these mass patterns are allowed by all the data at present. Therst two alternatives give a contribution to effective neutrino massprobed in neutrinoless double beta decay which is at most of theorder of the atmospheric scale while the last possibility can give

a contribution which is close to the present experimental limit.It is expected [19] that the future neutrinoless double beta decayexperiments and direct measurement of the electron neutrino masswill help in distinguishing between the above alternatives.

The mixing pattern among neutrinos is also strongly con-strained by the available results. To discuss this, we need togeneralize the two generation oscillation picture to the realisticcase of three neutrinos. The neutrino production is controlled bythe following charged current reaction:

Lch =g

2 2(lL U i iL W + h.c. ). (1.50)

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Here U is a unitary matrix describing neutrino mixing. If CPconservation is assumed then U can be parameterized in terms of three mixing angles:

U =cc sc scss + cs cc sss sc

ccs + ss ssc + sc cc, (1.51)

A neutrino beam of a denite avour evolves in vacuum at timet to

| (t) = U i eiE i t | i .The probability P that this beam contains avour after trav-eling distance L can be derived as in Eq.(1.45):

P = 4i

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1.6. Models of neutrino masses 31

sin2 2 0.8 1,sin2 0.01. (1.54)

One particular example of the mixing matrix which describesthe experimental results is obtained by taking s = 0 and = = 4 :

U =

1 2 1 2 012

12 1 2

12 1 2 1 2, (1.55)

We now discuss theoretical ways which can lead to the requiredmasses and mixing patterns among the three neutrinos discussedabove.

1.6 Models of neutrino massesThere is an extensive literature [25] on how to theoretically realizethe mass patterns discussed in the last section. We briey discussthe ideas involved. All the mechanisms for neutrino mass genera-tions nally lead to an effective mass term for the light neutrinosdened as follows:

Lm =12

iL m ij c jL + h.c. (1.56)

Here i, j are generation indices and we assumed only three lightneutrinos. m is a complex symmetric 3

3 matrix.

The operator written above is not SU (2) invariant and couldarise only after spontaneous breaking of this symmetry. Moreover,it transforms as an SU (2)-triplet. Thus one needs to generate aneffective SU (2)-triplet Higgs eld. There are various ways of doingthis. All of these need extension in the Higgs or fermion sector of the standard model. The possibilities are:

1. Add one or more singlet fermions N : The mixing of thestandard neutrino with N along with the lepton numberviolating mass term for N generates m in Eq.(1.56). This isthe seesaw mechanism already encountered.

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2. Add a triplet Higgs eld : Direct coupling of triplet with lep-tonic doublet leads to Eq.(1.56). The triplet vacuum expec-tation value determines neutrino mass scale and is required

to be small. This gives so called triplet majoron model[30]if lepton number is spontaneously broken. Consistency of this model with Z width requires explicit violation of leptonnumber or addition of a singlet Higgs [31] also.

3. Add a singly or doubly charged Higgs eld : Coupling of thisadditional Higgs eld to the left or the right handed leptonsradiatively generate Eq.(1.56). This scenario is realized inthe Zee [32] and the Zee-Babu [33] models.

Various neutrino mass models are derived using one or more of the above ingredients. These models have to answer two basic

questions. (1) Why neutrino masses are much smaller compared toother fermion masses and (2) why they mix so strongly leading totwo large mixing angles. To see how these questions are answered,we discuss two of the above possibilities in some detail.

1.6.1 Seesaw models

The neutrino mass term in this model is analogous to Eq.(1.19)but now for three left-handed and three right handed neutrinosN iR :

Lmass = iL (mD )ij N jR +12[(N iR )

c(M R )ij N jR + h .c.], (1.57)

The above equation leads to the following mass matrix

M =0 mD

mT D M R. (1.58)

Both mD and M R are matrices in generation space. When M R isnonsingular and is given by a scale M much larger than that inmD , we get

m mD M 1R mT D . (1.59)

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1.6. Models of neutrino masses 33

All the neutrino masses are automatically suppressed due to alarge scale M in M R . A large M is natural in grand unied SO (10)theories which therefore provide a nice framework to understand

small neutrino masses. One gets the following mass hierarchy fora diagonal M R

m1 : m2 : m3 :: m2D 1 : m2D 2 : m

2D 3,

where mDi are eigenvalues of mD . As long as these eigenvaluesare hierarchical, the neutrino masses also display the hierarchy.There are however ways to obtain non-hierarchical masses as wellas large mixing angles [25, 34] within the seesaw models.

1.6.2 Radiative models

It can happen that some symmetry ( e.g. lepton number) for-bids neutrino mass term at the tree level even after extending theStandard Model elds. Soft breaking of this symmetry may how-ever radiatively induce a nite neutrino mass at one or two looplevel. This provides an attractive mechanism for understandingthe smallness of neutrino masses. The radiatively induced contri-bution to neutrino masses may be present in addition to the treelevel mass. The presence of radiative contribution in this casemay explain the hierarchy among neutrino masses. This happensin case of the supersymmetric theories containing explicit leptonnumber violating interactions.

Radiative mechanism for neutrino mass generation can be im-plemented in a number of ways. A nice classication of all thesepossibility can be found in [35]. Many of these models are varia-tions of the basics mechanism proposed by Zee[32]. This mecha-nism needs an extended Higgs sector containing a charged singleteld h+ and a doublet eld 2 in addition to the standard Higgsdoublet 1. This allows the following additional terms in the SMLagrangian:

LZee = f lc l h+ + h.c.,V = 12h+ + h.c. (1.60)

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The rst equation by itself cannot generate neutrino mass at treelevel since it does not contain a bilinear term in neutrino elds. Itcannot give rise to radiative mass also since this term conserves

lepton number with appropriately dened lepton number for theeld h+ . It is not possible to dene a conserved lepton num-ber when both the terms displayed above and the charged leptonYukawa couplings are simultaneously present. In the presence of all these terms, neutrinos obtain radiative masses.

The general structure of neutrino mass matrix obtained in thismodel is given by

m Zee =0 m e m e

me 0 m me m 0

, (1.61)

where m = Cf (m2 m2 )and C is a constant which depends upon the Higgs masses andmixing.

The consequences of this mass matrix have been extensivelystudied [36, 37, 38]. If one of the parameters m is suppressedcompared to the other two then the above mass matrix has ap-proximate symmetry of the type L -L -L . The above mass ma-trix then leads to a pseudo-Dirac and an almost massless neutrinodesired for phenomenology. But not all such cases can give thecorrect mixing pattern. For example if f are of similar mag-

nitudes then m e m

m e . This case leads to the expla-nation [37] of the atmospheric neutrino decit but cannot solvethe solar neutrino problem. By allowing hierarchy in f , it ispossible to obtain bi-large pattern and simultaneous solutions toboth anomalies [39, 36] but solar mixing angle stays very close tomaximal [39, 38] in contrast to the non-maximal mixing requiredin the LMA solution.

A variation of the above model is obtained by replacing h+with a doubly charged eld ++ [32, 33]. Such model does notneed addition eld 2 and lead to neutrino masses at two looplevel.

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1.7. Summary 35

We discussed above two of the most popular models of neu-trino masses. While these show typical theoretical ways to obtainneutrino masses, details can substantially change depending upon

the phenomenological requirements and theoretical prejudices. Adetailed discussion of these topics would take us out of the scopeof these lectures.

1.7 Summary

We discussed the subject of neutrino masses in this mini review.Three basic topics were covered: (1) the formalism of neutrinomasses (2) experimental methods to detect them and (3) dis-cussion of the presently allowed patterns and models of neutrinomasses. We had to completely leave out cosmological implications

of neutrino masses. Our main focus was a basic introduction tothe subject aimed at graduate students not familiar with it. Thematerial presented is incomplete and one would need to consultexisting books and reviews to ll the gaps. Our hope is that thebackground presented here would be sufficient to motivate andenable fresh students to understand the more elaborate detailspresented in these books and reviews.

Note Added: After these notes were written, the KamLand ex-periment in Japan obtained for the rst time positive evidence forthe oscillations of the man made neutrinos. This detector uses theanti neutrino uxes from 16 different nuclear reactors located inJapan and Korea. The average distance of these reactors is

180

km and average energy is around 3.0 MeV. This allows the detec-tor to look for the (mass) 2 differences1.610

5 eV2. The resultsfrom these detectors have conrmed the large mixing angle solu-tion to the solar neutrino problem. This experiment alongwith thelatest results from SNO now narrows down the allowed regions in(mass) 2 difference and mixing angle, for details, see the Plenarytalk at PASCOS03 by S. Goswami [40].

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Bibliography

[1] E.C.G. Sudarshan and R. Marshak, Phys. Rev. 109 1860

(1958).[2] R.P. Feynman and M. Gell-Mann, Phys. Rev. 109 193 (1958).

[3] E.D. Commins and P.H. Bucksbum, Weak interactions of lep-tons and quarks , Cambridge University Press (1983).

[4] B. Kayser, G. Debu and F. Perrier, The physics of massive neutrinos , World Scientic (1989).

[5] R.N. Mohapatra and P.B. Pal, Massive neutrinos in physics and astrophysics , World Scientic (1991).

[6] C.W. Kim and A. Pevsner, Neutrinos in physics and astro-physics , Harwood Academic Publishers (1993)

[7] E.S. Abers, and B.W. Lee, Phys. Rep. C9(1) 1,(1973);S. Weinberg, Rev. Mod. Phys. 46 (1974) 255.

[8] C.L. Cowan et al , Science 124 103 (1956).

[9] K. Kodama et al , Phys. Lett. B504 218 (2001).

[10] J. Bahcall et al , Solar neutrinos: The rst thirty years ,Perseus Publishing (1995).

[11] J. Bahcal, Neutrino astrophysics , Cambridge University Press(1989).

[12] P.H. Frampton and P. Vogel, Phys. Rep. 82 339 (1982).

[13] S.M. Bilenky and S.T. Petcov, Rev. Mod. Phys. 59 671(1987).

[14] T. K. Kuo and J. Pantaleone, Rev. Mod. Phys. 61 937 (1989).

[15] W.C. Hexton and B. Holstein, hep-ph/9905257.

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Bibliography 37

[16] A.S. Joshipura and S.D. Rindani, Phys. Lett. B494 (2000)114.

[17] A.S. Joshipura, hep-ph/0204305.[18] Particle Data Group, K. Hagiwara et al , Phys. Rev. D66

(2002) 010001.

[19] An incomplete list of reference is given byH. V. Klapdor-Kleingrothaus, H. Pas and A.Yu. Smirnov,Phys. Rev. D63 (2001) 73005;H. Minakata and O. Yasuda, hep-ph/9609276;W. Rodejohann, Nucl. Phys. B597 (2001) 110 and hep-ph/0203214;F. Vissani, hep-ph/9904349;

S.M. Bilenky et al , Phys. Lett. B465 (1999) 193;S.M. Bilenky, S. Pascoli and S.T. Petcov, Phys. Rev. D64(2001) 053010;H. Minakata and H. Sugiyama, hep-ph/0111269 and hep-ph/0202003;S. Pascoli and S.T. Petcov, hep-ph/0205022.

[20] The latest and earlier results and details of the experimentscan be found at www-sk.icrr.u-tokyo.ac.jp

[21] B.T. Cleveland et al , Astrophys. J. 496 (1998) 505.

[22] The Gallex collaboration, Phys. Lett. 447 (1999) 127.

[23] The SAGE collaboration, astro-ph/0204245.

[24] The SNO collaboration, nucl-ex/0204008 and nucl-ex/0204009. See also www.sno.phy.queensu.ca/sno .

[25] For reviews of various models, see,A.Yu. Smirnov, hep-ph/9901208;R.N. Mohapatra, hep-ph/9910365;A.S. Joshipura, Pramana 54 (2000) 119.

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[26] A. Bandyopadhyay et. al , hep-ph/00204286;V. Barger et al , hep-ph/0204253;J.N. Bahcall et. al , hep-ph/0204314;

P.C. de-Hollanda and A.Yu. Smirnov, hep-ph/0205241;C.V.K. Baba, D. Indumathi and M.V.N. Murthy, Phys. Rev.D65 (2002) 073033 .

[27] The LSND collaboration, hep-ex/0104049 and Phys. Rev.Lett. 81 (1998) 1774.

[28] The Karmen Collaboration, hep-ex/0203021 and E.D.Church et al , hep-ex/0203023.

[29] CHOOZ collaboration, M. Apollonio et al , Phys. Lett. B466(1999) 415.

[30] G.B. Gelmini and M. Roncadelli, Phys. Lett. 193 (1981) 297.[31] A.S. Joshipura, Int. J. Mod. Phys. 7 (1982) 2021.

[32] A. Zee, Phys. Lett. B93 (1980) 389.

[33] K.S. Babu, Phys. Lett. B203 (1988) 132.

[34] I. Dorsner and S.M. Barr, Nucl. Phys. B617 (2001) 493.

[35] K.S. Babu and E. Ma, Mod. Phys. Lett. A4 (1989) 1975.

[36] Y. Koide, Nucl. Phys. Proc. Suppl. 111 , 294 (2002) [hep-ph/0201250].

[37] A.Y. Smirnov and M. Tanimoto, Phys. Rev. D 55 , 1665(1997) [hep-ph/9604370].

[38] B. Brahmachari and A.S. Choubey, Phys. Lett. B531(192002) 99;

[39] A.S. Joshipura and P. Krastev, Phys. Rev. D50 (191994) 31.

[40] S. Goswami, Plenary talk at the International Symposium,PASCOS03, Bombay, India (2003), hep-ph/0307224.

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Chapter 2

Higgs Physics

Saurabh D. Rindani

2.1 Introduction

Higgs physics is at present poised at an interesting juncture, whena light Higgs boson of the Standard Model (henceforth to be re-ferred to as SM), a spin-zero particle which would signal spon-taneous gauge symmetry breaking in the simplest form, has notbeen seen until the conclusion of experiments at LEP and LEP2electron-positron collider at CERN, Geneva. It is possible that on-going experiments at the p p collider Tevatron at FNAL in U.S.A.may discover the SM Higgs boson if the mass is not too large. If

it is not seen at Tevatron, one will have to wait until results comeout of the LHC (Large Hadron Collider) which is being built atCERN for a heavier Higgs. From a theoretical point of view, thedevelopments until the present time are complex and interesting.While some of the basic principles underlying spontaneous sym-metry breaking of gauge symmetry and the Higgs mechanism arenow commonly known, the actual realization of this mechanismin nature is still a subject of investigation. The mass of the SMHiggs boson is an unknown parameter and the pheonemonologyis sensitively dependent on the mass. Thus the properties anddiscovery strategies for the Higgs vary greatly depending on the

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40 2. Higgs Physics

supposed mass, and the phenomenology rapidly gets complex asthe range of the Higgs mass is increased.

This work, based on four lectures delivered mainly for Ph.D.

students, aims at describing aspects of the SM Higgs. The scope islimited in extent and depth because of the limited time availablefor the lectures.

Note that in the following, the SM Higgs is usually denoted byH , but at places, the Higgs eld will be denoted by , or, in thecontext of the unitarity gauge, by .

Some useful general references are [1, 2, 3, 4, 5].

2.2 The Standard Model Higgs boson

A Higgs boson is a massive spin-0 particle appearing in a local

gauge theory, where the local gauge invariance is broken com-pletely, or at least partially, by the mechanism of spontaneoussymmetry breaking. The simplest way in which spontaneousbreaking of a symmetry is achieved is by the introduction of el-ementary scalar elds in a theory. The ideas in the context arebest illustrated by means of the Standard Model of electromag-netic, weak and strong interactions. Since SM is currently a the-ory which explains most experimentally observed phenomena withfairly high accuracy, it is an interesting starting point.

2.2.1 Introduction to the Standard Model

The Standard Model (SM) of electromagnetic, weak and stronginteractions is based on a gauge theory with an SU (2)L U (1) SU (3)C gauge group. The quarks and leptons transform accord-ing to left-handed (LH) doublet and right-handed (RH) singletrepresentations of SU (2)L to account for the V A nature of the charged-current weak interactions. The quarks (both LH andRH) transform as triplets of SU (3)C of colour, to account for thestrong interactions of the quarks. The leptons are singlets underSU (3)C . The assignment of weak hypercharge corresponding tothe U (1) group to the various SU (2)L and SU (3)C multiplets is

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2.2. The Standard Model Higgs boson 41

according to the charge formula:

Q = T 3L + Y, (2.1)

where Q is the electric charge, T 3L the third component of weakisospin corresponding to SU (2)L , and Y the weak hypercharge.Since no generator of SU (3)C appears in eq. (2.1), electric chargeis independent of colour.

Quarks and leptons come in three generations:

L i iei L

; Q i u idi L

; i = 1 , 2, 3 = 1 , 2, 3eiR ; u iR ; di R .

(2.2)

Here i is the generation index, and the colour index.

The fermions interact with the gauge bosons, which correspondto the adjoint representation of the gauge group, through the min-imal coupling:

Lfermion = L iD/ L + R iD/ R , (2.3)where the sum is over all fermion multiplets, quarks as well asleptons. D is the covariant derivative appropriate to the repre-sentation:

D + ig2

a W a + ig6

B + igs2

a Ga (2.4)

for left-handed quarks,

D + ig2

a W a ig2

B (2.5)

for left-handed leptons,

D + ig QqB + igs2

a Ga (2.6)

for a right-handed quark q of charge Qq, and

D ig B (2.7)

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42 2. Higgs Physics

for a right-handed electron with charge 1. The notation is thatD is a differential operator and a matrix in SU (2)L space (2 2)as well as in SU (3)C colour space (3 3).The covariant derivative guarantees that Lfermion is invariantunder local gauge transformations

U, (2.8)where, for LH quarks, for example,

U = U 1U 2U 3, (2.9)

with

U 1 = eig Y Q 1 (x) ; U 2 = eig2

a a2 (x) ; U 3 = eig s2

a a3 (x) , (2.10)

and similarly for other elds. The gauge elds transform as

ig2

a W a U 2 U 12 + U 2 ig2

a W a U 12 , (2.11)

igs2

a Ga U 3 U 13 + U 3 igs2

a Ga U 13 , (2.12)

ig B ig 1 + ig B. (2.13)The Lagrangian for gauge elds may be written as

Lgauge =

1

2B B

1

2W a W

a

1

2Ga G

a , (2.14)

whereB = B B , (2.15)

ig2

a W a = ig2

a W a W a + [ ig2

a W a , ig2

bW b ], (2.16)

igs2

a Ga = igs2

a Ga Ga + [ igs2

a Ga , igs2

bGb ]. (2.17)

The last two may be rewritten as

W a = W a W a g abc W bW c , (2.18)

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Ga = Ga p Ga gs f abcGbGc , (2.19)

where abc is the Levi-Civita symbol and f abc are the SU (3) struc-

ture constants.In the gauge-invariant theory described above, none of theelds correspond to massive particles. Fermion mass terms areof the the form m L R + h .c., and are forbidden by even globalgauge invariance. Gauge boson mass terms are forbidden by lo-cal gauge invariance. The way to give masses without sacricingthe renormalizability of the theory is by allowing for spontaneoussymmetry breaking.

2.2.2 Spontaneous symmetry breaking

A symmetry is said to be broken spontaneously if the Lagrangian

of the theory is invariant under the symmetry, but the groundstate (the vacuum state) is not invariant. In this situation, manydegenerate vacuua are related to one another by the symmetry of the Lagrangian, and one is singled out to be the correct vacuum.Hence states constructed out of this vacuum reect this bias, andthe dynamics of the theory no longer shows the invariance.

The Lagrangian we described has a unique minimum energystate corresponding to all the elds taking the value zero. Thisvacuum is gauge invariant. Hence something has to be added tothe theory to achieve spontaneous symmetry breaking.

Moreover, if the vacuum expectation value of any but a spin-0

eld is nonzero, even Lorentz invariance would be violated. Toavoid this, one introduces only scalar elds in the theory. Thescalar eld(s) may be elementary or composite. The latter case isnot impossible in the standard model as such, but requires extrainteractions which can generate a composite scalar eld corre-sponding to a bound state.

Let us look more closely at the phenomenon of spontaneoussymmetry breaking with elementary scalar elds.

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44 2. Higgs Physics

A. Abelian global symmetryWe start with the simple example of a theory with a sin-

gle complex scalar eld with a global U (1) symmetry under

(x) ei (x), where is independent of space and time. Thecorresponding Lagrangian may be written asL(, ) = V (). (2.20)

We can write the potential term as

V () = 2 + ()2, (2.21)

where higher order terms in are not included because theywould destroy the renormalizability of the theory.

The Hamiltonian density corresponding to Lis

H=.. + + V (). (2.22)

Since the rst two terms are positive (semi-)denite, it is clearthat the eld conguration which minimizes the HamiltonianH = Hd3x should be space and time independent, and shouldminimize V (). For 2, > 0, the only minimum of the poten-tial V is = 0. This will correspond to a vacuum state which issymmetric, with no U (1) charge. However, > 0, 2 < 0 givesrise to an additional set of nontrivial minima, at

=

2

2, (2.23)

that is, at = 2/ (2)ei , for 0 < < 2. These congu-rations are not only local minima, but also global minima, withthe energy density 4/ (4). Note that > 0 ensures stability.Since for this state to make sense it has to be unique, has to takesome xed value. Any value of corresponds to an unsymmetricvacuum.

Let us work with a shifted eld which has a zero value for thisminimum:

(x) = (x) v 2 , (2.24)

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where v = 2/ , and where we have chosen = 0 for conve-nience. Then the Lagrangian of eq. (2.20) becomesL( , ) =

2 +

v

2

+v

2

+v 2

+ v 22

. (2.25)

On simplication, this gives

L( , ) = ( )2 + 2 2(Re )2+ 2 2v( )Re + 4

4. (2.26)

If we write the above equation with

= 1 2(1 + i2), (2.27)where 1,2 are real, the kinetic energy terms have the correct nor-malizations:

L=12

1 1+12

2 2+ 221+ v31+ 1

22

4

21 + 22

2.

(2.28)It can be seen that 1 has a mass term, with mass 22, whereasthe eld 2 is massless. The latter corresponds to the masslessboson predicted by the Goldstones theorem for spontaneouslybroken global continuous symmetries [6].

B. Abelian local gauge symmetryLet us consider the same U (1) symmetry, but now local. We

again take our simple scalar model. Then, gauge invariance re-quires the use of a covariant derivative D + igA , where Ais the gauge eld. The gauge-invariant Lagrangian now is

L= DD V () 14

F F . (2.29)

It is invariant under ei (x), (2.30)

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46 2. Higgs Physics

A A 1g

. (2.31)

A trivial vacuum corresponds to = 0, A = 0, which is the

case for 2

= 0. For 2

< 0, again, the vacuum corresponds to = v/ 2, with v2 = 2/ . We again make a shift(x) =

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