Fundamentals of Neutrino Physics andAstrophysics

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Fundamentals of Neutrino Physics

and Astrophysics

Carlo GiuntiIstituto Nazionale di Fisica Nucleare, Sezione di Torino and Dipartimento di

Fisica Teorica, Universita di Torino, Italy

Chung W. KimKorea Institute for Advanced Study, Seoul, Korea and The Johns Hopkins

University, Baltimore, MD, USA

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Dedicated to the late Young J. Kim, beloved wife of C.W.K. and dear friend of

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PREFACE

An expert is a person who avoids thesmall errors while sweeping on to thegrand fallacy.Steven Weinberg

Studying the properties and interactions of neutrinos has been one of the mostexciting and vigorous activities in particle physics and astrophysics ever since Paulifirst proposed their existence in 1930. In spite of their weakly interacting (or barelyexisting, as Lederman used to say) nature, we have so far accumulated an enormousamount of knowledge about neutrinos. From neutrino oscillation experiments, welearned a few years ago that neutrinos are massive and mixed. However, we still donot know the absolute values of their masses and some aspects of the mixing.

In this book we have tried to gather all the basic knowledge and tools thatare necessary to understand and to infer the true nature of the neutrinos fromthe experimental data, using the theories that have been developed. We have alsosummarized the well-established facts concerning neutrinos and the important roleplayed by neutrinos in the Sun and supernovae, and in shaping the Universe we livein. Special emphases are placed on the basic knowledge of how neutrinos interact,how they behave in matter as well as in vacuum, and on the formal aspects ofthe theory of neutrino oscillations. Salient features of the oscillation experimentsin various settings and sources are given with careful analysis.

After a short history of the neutrino, leading up to the Standard Model (SM)of electroweak interactions and the discovery of neutrino oscillations in chapter 1,chapter 2 is devoted to a detailed discussion of the properties of spin 1/2 Diracparticles. This chapter can be used as an introduction to the quantum field theoryof spin 1/2 fermions. We have tried to make chapter 2 as complete and self-containedas possible, especially for beginners, by including some details on the terminologyused in quantum field theory and in gauge theory. The symmetry properties of spin1/2 particles under charge conjugation, C, parity, P, and time reversal, T, as wellas the space-time and Lorentz transformations are discussed. The realistic wavepacket description of a particle is also presented.

In chapter 3, the ingredients necessary for understanding the Standard Model,such as gauge symmetry and the Higgs mechanism, are presented. The electroweakbehavior of quarks and leptons is explained and summarized, together with somediscussions of the gauge bosons involved.

One of the most remarkable discoveries in the past decade is the finding thatneutrinos are massive and mixed. Chapter 4 is devoted to a detailed discussionof the three-generation mixing of quarks, which can be extended to the treatmentof mixing of three Dirac neutrinos in a straightforward way (the mixing of threeMajorana neutrinos implies the existence of two additional phases in the mixingmatrix, as explained in chapter 6). A construction of the mixing matrix is presented

viii PREFACE

with mathematical details. The possibility of CP violation due to mixing and howto describe and quantify it are explained.

Chapter 5 is devoted to neutrino interactions, with discussions of several impor-tant processes involving neutrinos, ranging from neutron decay to charged-currentand neutral-current deep inelastic neutrino–nucleon scattering. All the discussionsare based on the results obtained in chapter 3, where the SM was developed.

We present, in chapter 6, the description of the general, model-independent,theory of massive neutrinos. Depending on whether they are of Dirac or Majoranatype, the mathematical description takes different forms. In addition, we discuss thefamous see-saw mechanism, which explains the smallness of the neutrino masses ina natural way. Because of their speculative nature, we have intentionally omitteddetails concerning the possible neutrino mass-generating mechanisms that havebeen put forward by many authors.

Equipped with the mathematical and basic physics background in the previouschapters, we present, in chapter 7, the standard derivation of the neutrino oscil-lation probability. We have tried to make this chapter as complete as possible inthe hope that the reader can find all the necessary information on the derivationof the standard oscillation formulas, together with how to use them in analyz-ing experimental data. The types of experiments are as numerous as they can be:experiments with energetic and low energy neutrinos, with reactor or acceleratorneutrinos, solar, atmospheric, and even extragalactic (SN 1987A) neutrinos, andfinally with short-baseline and long-baseline arrangements. We have included help-ful discussions on all the possible cases. In addition, some consequences of C, CP, Tand CPT violation for the oscillations, and special cases of oscillations with differentoscillation parameters are discussed.

In chapter 8, in view of the important role that the oscillation plays in probingthe neutrino properties, a derivation of the neutrino oscillation formulas is givenwith an emphasis on the more realistic relativistic wave packet treatment. Alsopresented in this chapter are answers to often raised questions concerning subtleand confusing issues in the oscillation formulas.

One of the most interesting findings in neutrino physics is the discovery thatthe properties of neutrinos change when they pass through dense matter. Due tothe difference of the weak interaction potentials that different flavor neutrinos feelin matter, the effective mixing can be dramatically enhanced in the case of a verysmall mixing in vacuum. If the neutrinos pass through a region in which the effectivemixing is maximal, it is possible to have large flavor transitions. In chapter 9, weexplain this resonant effect, known as the MSW effect. We derive the necessaryformulas and explained the methods for analyzing the data in the simplest caseof two-neutrino mixing (three-neutrino mixing is discussed in chapter 13). Alsopresented is a geometrical description of the oscillations in vacuum as well as inmatter, which, we hope, will help the reader to understand better this importantphenomenon. The geometrical description shows that the neutrino oscillation isanalogous to a classical magnetic moment precessing in an external magnetic field.

By far, the longest running neutrino experiment has been the Homestake solarneutrino experiment, started in the late 1960s by Ray Davis and collaborators afterJohn Bahcall predicted a measurable rate on the basis of a solar model. In the early

PREFACE ix

1970s, Davis and collaborators discovered the so-called solar neutrino problem,which has remained unsolved until recent times. It consists of a solar neutrinodetection rate which is substantially smaller than that predicted by standard solarmodels. In chapter 10, we start with a brief explanation of the standard solarmodels, which explain how the Sun shines and how neutrinos are produced. Then,we list and describe all the experiments that have so far been performed, as well astheir results and their roles in the solar neutrino problem. We end this chapter bypresenting the results of a global fit of all the solar neutrino data. The results of thecombined analysis of the data of solar neutrino experiments and of the KamLANDvery long-baseline reactor neutrino experiment is presented later in chapter 12.

Chapter 11 explains the generating mechanism and the flux of atmosphericneutrinos. A comprehensive survey of all the atmospheric neutrino experimentsperformed so far is presented. It is interesting to note that the first undisputeddiscovery of neutrino oscillations was made with atmospheric neutrinos which havebeen initially considered as unwanted background for other measurements. Theconfirmation of this discovery by the accelerator long-baseline K2K experiment isdiscussed later in chapter 12.

In chapter 12 we start with the introduction of the sensitivity of terrestrialreactor and accelerator neutrino experiments to the measurement of the oscillationparameters. Then, we present the main results of reactor and accelerator short-baseline, long-baseline and very long-baseline experiments, with specific reviews ofthe CHOOZ, Palo Verde, KamLAND, and K2K experiments, which are importantfor our present knowledge of the neutrino oscillation parameters. This chapter endswith a brief discussion of the future off-axis long-baseline experiments.

Deciphering the values of the squared-mass differences and the mixing anglesfrom the data requires a careful analysis with use of the three-generation mixingmatrix, which is often quite complex. Technical issues involved in the analysis andsome useful approximations are discussed in chapter 13. Also discussed in thischapter are the results of a global analysis of all the existing oscillation data. Thechapter ends with some comments on the absolute scale of neutrino masses.

The most important attempts to measure directly the values of the neutrinomasses are the measurements of the end-point of the electron spectrum in nuclearβ-decays, in particular that of tritium, and neutrinoless double-β-decays. In chap-ter 14, we review the current experimental upper bounds on the neutrino massesobtained with tritium β-decay experiments with the effects of neutrino mixingstaken into account. Also given is a brief summary of neutrinoless double-β-decaytheory and experiments. We emphasize the salience of neutrinoless double-β-decayexperiments in determining whether the neutrino is a Dirac or a Majorana particle.Some useful comments on the results of the experiments are also given.

Chapter 15 is devoted to supernova neutrinos. In order to help the reader whois not familiar with the subject, a short introduction to supernova physics is pre-sented, explaining the types of supernovae and their explosion mechanisms. Thesupernova which is of special interest to us, and so best studied, is SN1987A. Theneutrino burst of SN1987A was detected by three experiments: Kamiokande-II,IMB, and Baksan. We present the data and compare them with theory. The limits

x PREFACE

on the neutrino masses are discussed, taking into account neutrino mixing, and theconstraints on other neutrino properties are briefly summarized.

In chapter 16 we present a brief introduction to the main aspects of the stan-dard cosmological model, which are necessary for understanding the relic neutrinos.These are the most abundant known, but not yet detected, relic particles in theUniverse, next to the Cosmic Microwave Background Radiation (CMBR). After theintroduction of the Standard Cosmological Model and the discussion of the dynam-ics of the expansion of the Universe, the thermodynamics of the early Universe andthe decoupling of relic particles are explained. Finally, we present the main prop-erties of the CMBR, which is one of the most important sources of cosmologicalinformation.

The final chapter 17 deals with the relic neutrinos. The decoupling of neutrinos,both light and possibly heavy ones, and the importance of neutrinos for nucleosyn-thesis in the early Universe are explained. Also discussed is the role of baryonic,cold, and hot dark matter in the formation of large-scale structures. A global fitof the cosmological data is presented to gain an insight into various limits on theneutrino masses, the number of neutrino species and the neutrino asymmetry inthe Universe.

We wish to emphasize that this book is not a revised version of the book Neu-trinos in Physics and Astrophysics, co-authored by one of the present authorsmore than 10 years ago. Here, by making the book as self-contained as pos-sible, we have presented all the necessary details for the reader to follow andunderstand the subject matter. This is why we have chosen the title Fundamen-tals of Neutrino Physics and Astrophysics. For curious and studious readers whowish to find even more details, we have included as many references as possi-ble, although no collection of references can be exhaustive. It is with our sincereregret that we have not discussed, in this book, some important topics, in partic-ular, the theories of neutrino masses, the electromagnetic properties of neutrinosand the phenomenology of high-energy neutrinos from astrophysical sources. Theinterested readers can find introductions and reviews of these three topics, respec-tively, in Refs. [812, 74, 467, 781, 1076, 673, 810, 75, 763, 811, 815, 950, 813],Refs. [884, 52, 226, 158] and Refs. [487, 499, 584, 979, 785, 590, 583].

ACKNOWLEDGEMENTS

C.W.K. wishes to thank many colleagues at the Johns Hopkins University and theKorea Institute for Advanced Study; in particular, Gordon Feldman, Jon Bagger,E.J. Chun, P. Ko, K. Lee, C.B. Park, H. Park and P. Yi for valuable discus-sions, and for continued interest and encouragement. C.G. would like to thank:all the colleagues at the Torino Section of INFN and the Physics Departmentof Torino University, in particular A. Bottino and N. Fornengo, for encourage-ment and illuminating discussions; S.M. Bilenky for many enlightening discussions,long collaboration and for sharing his knowledge and experience; M. Laveder forstimulating collaboration, for cooperation in the creation and development of theNeutrino Unbound web pages at http://www.nu.to.infn.it; M.V. Garzelli andM. Laveder for reading the manuscript, finding several weak points, and suggestingcorrections. Both of us are deeply indebted to the Korea Institute for AdvancedStudy for providing us with several opportunities to work together on this project.We should like to thank Sonke Adlung of the Oxford University Press for his patientand unfailing encouragement over the past few years.

http://www.nu.to.infn.it

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CONTENTS

1 Historical introduction 1

2 Quantized Dirac fields 72.1 Dirac equation 72.2 Representations of γ matrices 92.3 Products of γ matrices 112.4 Relativistic covariance 132.5 Helicity 172.6 Gauge transformations 172.7 Chirality 182.8 Solution of the Dirac equation 222.9 Quantization 312.10 Symmetry transformation of states 362.11 C, P, and T transformations 482.12 Wave packets 602.13 Finite normalization volume 632.14 Fierz transformations 64

3 The Standard Model 673.1 Electroweak Lagrangian 713.2 Electroweak interactions 753.3 Three generations 803.4 The Higgs mechanism 833.5 Fermion masses and mixing 883.6 Gauge bosons 973.7 Effective low-energy CC and NC Lagrangians 102

4 Three-generation mixing 1064.1 Diagonalization of the mass matrix 1074.2 Physical parameters in the mixing matrix 1084.3 Parameterization of the mixing matrix 1094.4 Degenerate masses 1164.5 Mixing matrix with one vanishing element 1184.6 CP violation 1204.7 Rephasing invariants 1244.8 Unitarity triangles 1294.9 Conditions for CP violation 133

5 Neutrino interactions 1355.1 Neutrino–electron interactions 136

xiv CONTENTS

5.2 Hadron decays 1475.3 Neutrino–nucleon scattering 160

6 Massive neutrinos 1806.1 Dirac masses 1806.2 Majorana neutrinos 1886.3 Mixing of three Majorana neutrinos 2086.4 One-generation Dirac–Majorana mass term 2166.5 Three-generation Dirac–Majorana mixing 2296.6 Special cases 2356.7 Majorana mass matrix 237

7 Neutrino oscillations in vacuum 2457.1 Standard Derivation of the Neutrino Oscillation Probability 2477.2 Antineutrino case 2547.3 CPT, CP, and T transformations 2567.4 Two-neutrino mixing 2597.5 Types of neutrino oscillation experiments 2617.6 Averaged transition probability 2677.7 Large ∆m2 dominance 2737.8 Active small ∆m2 277

8 Theory of neutrino oscillations in vacuum 2838.1 Plane-wave approximation 2848.2 Wave-packet treatment 2998.3 Size of neutrino wave packets 3118.4 Questions 316

9 Neutrino oscillations in matter 3229.1 Effective potentials in matter 3239.2 Evolution of neutrino flavors 3299.3 The MSW effect 3319.4 Slab approximation 3399.5 Parametric resonance 3419.6 Geometrical representation 343

10 Solar neutrinos 35210.1 Thermonuclear energy production 35310.2 Standard solar models 35910.3 Model-independent constraints on solar neutrino fluxes 36410.4 Homestake experiment 36610.5 Gallium experiments 36810.6 Water Cherenkov detectors 37210.7 Vacuum oscillations 38110.8 Resonant flavor transitions in the Sun 38210.9 Regeneration of solar νe’s in the Earth 387

CONTENTS xv

10.10 Global fit of solar neutrino data 389

11 Atmospheric neutrinos 39011.1 Flux of atmospheric neutrinos 39311.2 Atmospheric neutrino experiments 416

12 Terrestrial neutrino oscillation experiments 42812.1 Sensitivity 42912.2 Reactor experiments 43212.3 Accelerator experiments 443

13 Phenomenology of three-neutrino mixing 45213.1 Neutrino oscillations in vacuum 45313.2 Matter effects 46513.3 Analysis of oscillation data 474

14 Direct measurements of neutrino mass 48414.1 Beta decay 48514.2 Pion and tau decays 49314.3 Neutrinoless double-beta decay 494

15 Supernova neutrinos 51115.1 Supernova types 51215.2 Supernova rates 51515.3 Core-collapse supernova dynamics 51715.4 SN1987A 52815.5 Neutrino mass 53415.6 Neutrino mixing 53515.7 Other neutrino properties 53615.8 Future 537

16 Cosmology 54016.1 Basic general relativity 54016.2 Robertson–Walker metric 54316.3 Dynamics of expansion 55316.4 Matter-dominated Universe 56016.5 Radiation-dominated Universe 56216.6 Curvature-dominated Universe 56316.7 Vacuum-dominated Universe 56316.8 Thermodynamics of the early Universe 56416.9 Entropy 56916.10 Decoupling 57216.11 Cosmic microwave background radiation 577

17 Relic neutrinos 58617.1 Neutrino decoupling 587

xvi CONTENTS

17.2 Electron-positron annihilation 58817.3 Neutrino temperature 58917.4 Energy density of light massive neutrinos 59017.5 Energy density of heavy neutrinos 59117.6 Big-Bang nucleosynthesis 59617.7 Large-scale structure formation 60017.8 Global fits of cosmological data 61217.9 Number of neutrinos 61817.10 Neutrino asymmetry 621

Appendices

A Conventions, useful formulas, and physical constants 626A.1 Conventions 626A.2 Pauli matrices 628A.3 Dirac matrices 629A.4 Mathematical formulas 634A.5 Physical constants 635

B Special relativity 637B.1 The Lorentz group 637B.2 Representations of the Lorentz group 643B.3 The Poincaré group and its representations 646

C Lagrangian theory 649C.1 Variational principle and field equations 649C.2 Canonical quantization 650C.3 Noether’s theorem 650C.4 Space-time translations 652C.5 Lorentz transformations 653C.6 Complex fields 653C.7 Global gauge symmetry 654

D Gauge theories 657D.1 General formulation of gauge theories 657D.2 Quantum chromodynamics 662

E Feynman rules of the standard electroweak model 664E.1 External lines 664E.2 Internal lines 665E.3 Vertices 666E.4 Cross-sections and decay rates 668

Bibliography 671

Index 705

1

HISTORICAL INTRODUCTION

I have done something very bad today by proposing a particlethat cannot be detected; it is something no theorist shouldever do.Wolfgang Pauli

The history of weak interactions dates back to 1896, when Becquerel discovered theradioactivity of uranium. Three years later, Rutherford discovered that there weretwo different by-products, α and β, γ being discovered later. In 1914, Chadwickdemonstrated that the β-spectrum was continuous, in contrast to α- and γ-rayswhich were unique in energy. This surprising result was subsequently confirmedin 1927 by Ellis and Wooster. Meitner later demonstrated that the missing energycould not be ascribed to neutral γ-rays, which led to the idea that the missing energycould be explained by the existence of a new particle or, as N. Bohr suggested,perhaps energy conservation held only in a statistical sense.

In order to remedy this serious problem as well as the problem of spin statisticsin β-decay, W. Pauli proposed, in an open letter to a physics conference at Tubingenon 4 December 1930, addressed to “Dear Radioactive Ladies and Gentlemen”, thatthe existence of a neutral weakly interacting fermion emitted in β-decay could solvethe problems (see Ref. [855]). He called this neutral fermion a neutron, with massof the order of the electron. In June 1931, Pauli gave a talk at a meeting of theAmerican Physical Society in Pasadena and reported for the first time on his idea.He did not have his talk printed, however, since he was still uncertain about hisidea.

When J. Chadwick discovered in 1932 the neutron as we know it today [310],E. Fermi renamed the Pauli particle the neutrino. The first published reference tothe neutrino is in the Proceedings of the Solvay Conference of October 1933. Fermi[430] and Perrin [865] independently concluded in 1933 that neutrinos could bemassless!

The first milestone in the theory of weak interactions was established in 1934when Fermi formulated a theory of β-decay [432, 431], now known as Fermi theory,in analogy with quantum electrodynamics (QED). In order to explain the observedchange of one unit of the nuclear spin in some β-decays, G. Gamow and E. Teller in1936 [505] extended the theory by introducing axial-vector currents in such a waythat parity was still conserved, since parity violation at that time was unthinkable.

It was then realized that other couplings such as scalar, pseudoscalar, and tensorcouplings could also participate in weak interactions. Due to this complication, forabout two decades the real combination of the couplings was in a state of extremeconfusion, in part, due to some erroneous experiments. A famous review article

2 HISTORICAL INTRODUCTION

by E.J. Konopinski concluded in 1955 [693], just before the discovery of parityviolation, that the correct form was a combination of scalar and tensor couplings,strongly biased by an impressive but wrong experiment with 6He.

The horizon of the weak interactions was further extended by the discovery of themuon, µ, in 1937, by J.C. Street and E.C. Stevenson [989] and S.H. Neddermeyerand C.D. Anderson [825]. The observations of muon decay led B. Pontecorvo topropose in 1947 [879] the universality of the Fermi interactions of electrons andmuons. This universality was further discussed by G. Puppi [885], O. Klein [682],J. Tiomno and J.A. Wheeler [1017] and T.D. Lee, M. Rosenbluth and C.N. Yang[726]. This may be the origin of the concept now known as generation or family.

Although the remarkable success of the Fermi theory left few in doubt of theneutrino’s existence, none had yet been observed in interactions, partly because ofthe predicted strength of interactions by H. Bethe and R. Peierls, who claimed in1934 that it might never be observed [220]. Urged, in particular by B. Pontecorvo inthe early 1950s, F. Reines and C.L. Cowan searched for a way to measure inverse β-decay, in which an antineutrino can produce a positron. After considering severalmethods, including a nuclear explosion, they settled on using the large flux ofantineutrinos from a nuclear reactor and 10 ton of equipment, including 1400 litersof liquid scintillators. This experiment was the first reactor-neutrino experiment. InJune of 1956, Reines and Cowan sent a telegram informing Pauli of the discovery[899] (see Ref. [897]). Reines (Cowan passed away) was awarded the Nobel prize 40years later!

First indicated in cosmic ray experiments and later confirmed by precise accel-erator experiments, K+ was found to decay into two different modes with oppositeparity. This was the famous θ-τ puzzle; K+, the one called θ, decays into two pions,whereas K+, the one called τ , decays into three pions. The puzzle was that θ and τhave the same mass, spin, and charge, i.e. they are the same particle! This cannothappen if parity is conserved in weak interactions.

Some started to question the validity of parity conservation, but it was T.D. Leeand C.N. Yang who first noted in 1956 [728] that evidence for parity conservationin weak interactions was lacking, not just in K decays but in all observed weakinteractions in the past. A number of tests to observe parity violation were suggestedby Lee and Yang. Subsequently, parity violation was observed in the β-decay ofpolarized 60Co [1073], π+ → µ+ + νµ and µ+ → e+ + νe + ν̄µ.

Once parity violation is allowed, the weak Lagrangian becomes even more com-plicated due to the appearance of parity violating couplings as well as the conservingones. However, this seemingly confused situation was dramatically simplified in theform of the V − A theory. The structure of the V − A theory, formulated in 1958by R.P. Feynman and M. Gell-Mann [434], E.C.G. Sudarshan and R.E. Marshak[993] and J.J. Sakurai [918] can easily be realized in the lepton sector by using thetwo-component theory of a massless neutrino, proposed in 1957 by L. Landau [711],T.D. Lee and C.N. Yang [727] and A. Salam [919]. (The idea was first developedby H. Weyl in 1929, but it was rejected by Pauli in 1933 on the grounds that itviolates parity [854].) In this theory, neutrinos are left-handed and antineutrinosare right-handed, leading automatically to the V −A couplings.

HISTORICAL INTRODUCTION 3

In 1958, Goldhaber, Grodzins and Sunyar [549] measured the polarization of aneutrino in the electron capture e− + 152Eu → 152Sm∗ + νe, with the subsequentdecay 152Sm∗ → 152Sm+γ. They found that the measured polarization of the pho-ton implies that the polarization of the νe was indeed in a direction opposite to itsmotion, within experimental uncertainties, in agreement with the two-componenttheory of a massless neutrino.

The concept of lepton number, L, was introduced in 1953 by E.J. Konopin-ski and H.M. Mahmoud to explain certain missing decay modes. The particlese−, µ−, τ−, νe, νµ, ντ are assigned with L = 1, whereas their antiparticles haveL = −1. In the V −A theory and today’s Standard Model of weak interactions, L isconserved. The Reines–Cowan experiment was consistent with lepton number con-servation. On the other hand, R. Davis’s attempts to observe ν̄e +

37Cl → 37Ar+e−turned out to produce only some limits on the cross-section for the process, becausethe process violates the lepton number. This effort, however, led Davis to thehistory-making Homestake solar neutrino experiment when he replaced ν̄e by νefrom the Sun.

Although lepton number conservation allows the reaction µ→ e+ γ, its experi-mental limits were many orders of magnitude smaller than predicted. This suggesteda new conservation law, one which assigns different lepton numbers to each leptonfamily, making µ → e + γ forbidden. This assignment led to the prediction thatνµ + n → p + e− is forbidden. As suggested by Pontecorvo [882], if it is shownthat νµ produced in π

+ → µ+ + νµ cannot induce e−, then νµ and νe are indeeddifferent particles. Encouraged by an estimate of the event rate by M. Schwartz[942], L.M. Lederman, M. Schwartz, J. Steinberger et al. [348] succeeded in 1962 atBrookhaven National Laboratory (BNL) in establishing the existence of the secondneutrino νµ. This experiment, which utilized an enormous amount of iron shieldingplates cut out of the battleship USS Missouri, marked the first serious acceleratorneutrino experiment.

A crucial milestone in the theory of weak interactions is the formulation ofthe Glashow–Weinberg–Salam Standard Model (SM) by S. Weinberg [1051] andA. Salam [920] in 1967. The model is based on an SU(2) × U(1) gauge modelproposed by S.L. Glashow in 1961 [543], which predicted the existence of weakneutral currents and the Z boson. The Standard Model incorporates the so-calledHiggs mechanism into the Glashow model. The Higgs mechanism, which was dis-covered in 1964 by P.W. Higgs [611, 610, 612], F. Englert and R. Brout [412],and G.S. Guralnik, C.R. Hagen and T.W.B. Kibble [578, 666], allows the originalmassless gauge bosons that appear in the local gauge group model to acquire lon-gitudinal degrees of freedom, finally making them massive as demanded in Nature.The renormalizability of the model was proved by G. ’t Hooft and M.J.G. Veltmanin 1971 [1002, 1001, 1003], elevating the model to an extremely viable one. Thesuccess of the SM was affirmed in 1973 by the discovery of neutral-current neutrinointeractions in the Gargamelle experiment at CERN (1973) [600, 599, 601] and wassubsequently confirmed at Fermilab [207].

The discovery in 1974 of the charm quark in the form of the J/ψ particle (cc̄) atBNL (J) [124] and SLAC (ψ) [126] and the subsequent discovery of W± [108, 160]and Z [109, 135] at CERN firmly established the SM as the model for leptonic and

4 HISTORICAL INTRODUCTION

hadronic weak and electromagnetic (electroweak) interactions. (The discovery ofthe charm quark was a triumph for the prediction of the existence of the charmquark as a result of the S.L. Glashow, J. Iliopoulos and L. Maiani (GIM) mechanism[544].) After the discovery of the third lepton, τ , by M. Perl in 1975 [862], the b andt quarks were discovered at Fermilab, respectively, in 1977 [608] and 1995 [22, 6],completing all the building blocks of the SM with three generations. Only the Higgsparticle, which is necessary in the mass-generating Higgs mechanism in the SM, hasnot yet been found. The number of generations was fixed at three in 1989 by theimpressive measurements by LEP experiments at CERN of the invisible width ofthe Z boson [362, 5, 29, 53].

In 1964 J.H. Christenson, J.W. Cronin, V.L. Fitch and R. Turlay [317] unex-pectedly discovered a violation of CP symmetry in K0-decay. The existence of thisviolation has been accommodated in the framework of the SM through the mixingof three generations of quarks by M. Kobayashi and T. Maskawa [685] in 1973,extending the theory of two-generation mixing developed by N. Cabibbo [292] in1963.

No experiments that have been performed so far have detected conclusive devia-tions from the SM, except neutrino oscillation experiments, which have shown thatneutrinos are massive and mixed. In the SM, this is not the case. This discoveryhas made the SM an effective theory of the yet unknown theory beyond the SM.The understanding of how the neutrinos would gain tiny masses and how they aremixed is an extremely challenging task that we have to face. The answer must befound in the theory beyond the SM. Thus, the neutrino is playing the role of amessenger of the new physics beyond the SM.

The concept of neutrino oscillations was first proposed in 1957 by Pontecorvo[880, 881], motivated by the K0 � K̄0 oscillation phenomenon (M. Gell-Mannand A. Pais [514], 1955), in which the strangeness quantum number is oscillating.The possible oscillations that he could find at that time were ν � ν̄ for Majo-rana neutrinos. Pontecorvo interpreted a rumor of Davis’s successful observation ofν̄+37Cl → 37Ar+e− with reactor antineutrinos (which turned out to be false [353])as a result of ν̄ → ν transitions and a subsequent ν + 37Cl → 37Ar + e− reaction.However, the V −A theory of weak interactions implies that, in the ultrarelativis-tic limit applicable to neutrinos, reactor antineutrinos are right-handed. Even ifthey oscillate into right-handed neutrinos (helicity is conserved), these neutrinoscannot induce the process ν + 37Cl → 37Ar + e−, which requires left-handed neu-trinos. A more realistic case of oscillations became available with the assumptionthat νe and νµ are mixed states of two mass eigenstates, which was discussed byZ. Maki, M. Nakagawa and S. Sakata in 1967 [766]. However, there was only a vaguehint of the present understanding of neutrino oscillations in their work. In 1967,Pontecorvo presented the first intuitive understanding of two-neutrino mixing andoscillations [883], which was later completed by V.N. Gribov and B. Pontecorvo in1969 [567]. The theory of neutrino oscillations was finally developed in 1975–76 byS. Eliezer and A.R. Swift [404], H. Fritzsch and P. Minkowski [466], S.M. Bilenkyand B. Pontecorvo [236, 239].

As in the case of many weak interaction experiments, neutrino oscillation exper-iments have had their own share of ups and downs in the early stages. The longest

HISTORICAL INTRODUCTION 5

running experiments by far have been those of the solar neutrinos. We have recentlywitnessed some spectacular results finally confirming the oscillations of the solarand atmospheric neutrinos and the reactor and accelerator neutrinos in varioussettings.

The atmospheric neutrinos, first regarded as unwanted background for theexperiments that had been designed to search for proton decay, have providedus with the first model-independent indication of oscillations of νµ’s through theSuper-Kamiokande experiment [476]. The atmospheric neutrino anomaly was dis-covered in the late 1980s in the Kamiokande and IMB experiments (see section 11.2).Nowadays, the high-precision measurements of the Super-Kamiokande experiment,confirmed by the measurements of the Soudan 2 and MACRO experiments (seesection 11.2), give us precise information on the values of the atmospheric neutrinooscillation parameters, which are in good agreement with the independent resultsof the first accelerator long-baseline K2K experiment (see section 12.3.2).

In the past decade, there have been many spectacular successes in the pur-suit of the solution of the solar neutrino problem, which, we believe, has finallybeen understood and solved to our satisfaction. This problem was discoveredin the Homestake experiment [323] and confirmed by the observations of theKamiokande, GALLEX/GNO, SAGE, Super-Kamiokande, and SNO experiments(see chapter 10). In particular, the results of the SNO experiment [43] have beeninstrumental in solving the solar neutrino problem in 2002. The depletion of thesolar neutrinos is finally found to be due to the oscillations of νe into νµ and ντinside the Sun by the Mikheev–Smirnov–Wolfenstein (MSW) resonance conver-sion effects [1065, 801, 802]. The solar model by the late John Bahcall and others[142, 137, 145, 152, 154] has become the Standard Solar Model which can now beused for the investigation of other solar properties. The reactor long-baseline Kam-LAND experiment has recently confirmed the values of the oscillation parametersobtained from a global analysis of the data of all solar neutrino experiments (seesection 12.2.3).

The results of the atmospheric, solar, KamLAND and K2K neutrino experi-ments are nicely explained by neutrino oscillations in the framework of the simplestmodel of three-neutrino mixing, in which the three flavor neutrinos νe, νµ and ντare unitary linear combinations of three massive neutrinos ν1, ν2, ν3 (see chap-ter 13). As of this writing, we have a rather precise knowledge of the value of theneutrino squared-mass difference ∆m221, the absolute value of ∆m

231 � ∆m232, and

the values of two mixing angles, ϑ12 and ϑ23. The value of the third mixing angle,ϑ13, the values of the CP phases (one for Dirac neutrinos and three for Majorananeutrinos), the absolute scale of neutrino masses, and the sign of ∆m231 remainunknown. On the value of ϑ13 we have only an upper bound, obtained from theabsence of neutrino oscillations observed in the reactor long-baseline experimentsCHOOZ and Palo Verde (see section 12.2.2). The absolute scale of neutrino massesis limited below a few eV by kinematical measurements of the electron spectrum inthe recent tritium β-decay experiments Mainz and Troitzk (see section 14.1).

An open question of fundamental importance is the nature of neutrinos, whichcould be either of Dirac or Majorana type. Neutrinoless double-β-decay experiments(see section 14.3) are considered the most promising way to decide if neutrinos

6 HISTORICAL INTRODUCTION

are Majorana particles. The current upper limit on the effective Majorana mass,obtained in the Heidelberg–Moscow 76Ge experiment [680], is of the order of oneeV, with an unfortunate uncertainty of a factor of about three due to the complexityof nuclear physics.

In recent years, we have learned that neutrinos play an important role in theearly Universe in many subtle ways. For example, the number of neutrino speciesaffects the primordial nucleosynthesis, which eventually decides the composition ofelements in the Universe [857, 981]. From this argument alone one could concludethat the number of neutrino species is close to three, in agreement with the precisenumber determined through the measurement of the invisible width of the Z-bosonby LEP experiments at CERN. Later in the evolution of the Universe, massiveneutrinos influence the formation of the large-scale structures in the form of hot darkmatter (see section 17.7.3). The recent observations (see chapters 16 and 17) of theCosmic Microwave Background Radiation (WMAP, Boomerang, and others), theLarge Scale Structures (2dFGRS and SDSS) and distant type Ia supernovae (High-z SN Search Team and Supernova Cosmology Project) and a precise determinationof the Hubble constant by the Hubble Space Telescope Key Project made possible abetter understanding of the evolution of the Universe. It turned out that only about5% of the energy of the Universe is composed of ordinary matter (baryons). Theother 95% is composed of invisible dark matter (∼ 25%) and mysterious dark energy(∼ 70%). Since the dark matter must be of the cold type (see section 17.7.2), hotdark matter in the form of massive neutrinos is severely constrained, leading to anupper bound on the sum of neutrino masses of the order of one eV (see section 17.8).This bound is remarkably close to the limits from totally independent experiments(tritium β-decay and neutrinoless double-β-decay experiments). The relic neutrinos,that decoupled from the rest of the primordial plasma when the Universe was aboutone second old, are the second most abundant particles in the Universe next to thephotons, with a number density smaller only by a factor of 3/11 for each family. Thedirect detection of the relic neutrinos still remains, however, as one of the biggestscientific challenges in the twenty first century, because of their weak interactions.

2

QUANTIZED DIRAC FIELDS

To trace the unfamiliar to the familiar is to understand.Algernon Blackwood, The Damned

In this chapter we review the main properties of quantized Dirac fields, whichdescribe particles with spin 1/2 (see, for example, Refs. [943, 634, 821]). In the Stan-dard Model, to be introduced in chapter 3, the fundamental fermions are quarks,charged leptons, and neutrinos, all of which have spin 1/2. In such a model, quarksand charged leptons are massive Dirac particles and neutrinos are massless Diracparticles. As discussed in chapters 10, 11, and 12, there is experimental evidencethat neutrinos are massive. In this case, neutrinos can be either Dirac particles orMajorana particles, as discussed in chapter 6. In any case, the Dirac theory pre-sented in this chapter represents the basis for the description of neutrinos, fromwhich the Majorana theory can be derived (see chapter 6).

2.1 Dirac equation

The Dirac Lagrangian for a free fermion field ψ(x) is1

L (x) = ψ(x) (i /↔∂ −m)ψ(x) , (2.1)

where ψ(x) is a spinor field with four components, the adjoint field ψ(x) being givenby

ψ(x) ≡ ψ†(x) γ0 , (2.2)

and↔∂µ ≡

→∂µ −

←∂µ

2, (2.3)

where→∂µ ≡ ∂µ is the normal derivative operator which acts on the right and

←∂µ is

a derivative operator which acts on the left (i.e. ψ←∂µ ≡ ∂µψ). For any four-vector

1 Often the Dirac Lagrangian is written as L ′(x) = ψ(x) (i/∂ −m)ψ(x), which differs

from eqn (2.1) by the total derivative L ′−L = i2∂µ

`ψ γµ ψ

´that has no effect on the field

equation derived from the Euler–Lagrange equation (C.9) (Gauss’ theorem implies thatthe integral of a total derivative is a surface term, which is invariant under the variationin eqn (C.3)). However, it is better to write the Dirac Lagrangian in the form eqn (2.1)because it is explicitly real, as a Lagrangian should be.

8 QUANTIZED DIRAC FIELDS

Aµ

/A ≡ γµAµ , (2.4)where γµ (µ = 0, 1, 2, 3) is a set of four 4×4 matrices2 called Dirac γ matrices thatsatisfy the anticommutation relations

{γµ, γν} ≡ γµγν + γνγµ = 2 gµν (2.5)

and the conditionγ0 㵆 γ0 = γµ . (2.6)

The γ matrices are constant in the sense that they do not transform underLorentz transformations. In other words, the γ matrices have the same value in allinertial reference frames. Although they are not four-vectors, it is convenient todefine formally the γ matrices with lowered indices as

γµ ≡ gµν γν (µ = 0, 1, 2, 3) ⇐⇒ γ0 = γ0 , γk = −γk (k = 1, 2, 3) . (2.7)

For µ �= ν the relations in eqn (2.5) imply that the four γ matrices anticommute.For µ = ν, the relations in eqn (2.5) constrain the squares of the γ matrices:

(γ0)2 = 1 , (γk)2 = −1 (k = 1, 2, 3) . (2.8)

The additional constraint in eqn (2.6) implies that

(γ0)† = γ0 , (γk)† = −γk ⇐⇒ (γµ)† = γµ . (2.9)

Thus, γ0 is Hermitian, whereas the γk matrices are anti-Hermitian.Using the Euler–Lagrange procedure (see eqn (C.9)), the field equations are

given by

∂µ∂L

∂(∂µψ)− ∂L

∂ψ= 0 , (2.10)

from which one obtains the Dirac equation

(i/∂ −m)ψ(x) = 0 . (2.11)

The anticommutation relations in eqn (2.5) are necessary in order to guaranteethe compatibility of the Dirac equation with the Klein–Gordon equation

(� +m2)ψ(x) = 0 , (2.12)

where � ≡ ∂µ∂µ. The Klein–Gordon equation must be satisfied by any free fieldbecause it is equivalent to the relativistic energy–momentum dispersion relation

2 The anticommutation relations in eqn (2.5) imply that the dimension N of the γmatrices is even. Indeed, let us consider the anticommutation relation with µ �= ν, thatis γµγν = −γνγµ = (−1) γνγµ. Taking the determinant, we obtain Detγµ Detγν =(−1)N Detγν Detγµ. Since Detγν �= Detγµ, we have (−1)N = 1, which implies that Nis even.

The minimal dimension of the γ matrices is N = 4, because there are only threeanticommuting matrices for N = 2, which are the Pauli matrices in eqn (A.29).

In practice the γ matrices are always represented in the minimal 4 × 4 dimensional-ity, because representations of higher dimensionality are equivalent but obviously morecomplicated.

REPRESENTATIONS OF γ MATRICES 9

in eqn (B.88). In fact, the Klein–Gordon equation has positive-energy plane-wavesolutions ψ(x) ∝ e−ip · x with p2 = m2. The negative-energy plane-wave solutionsψ(x) ∝ eip ·x are interpreted as antiparticle states, as explained in section 2.9.

The Klein–Gordon equation (2.12) is obtained from the Dirac equation (2.11)by multiplying it on the left by (i/∂ +m) and using the identity /∂ /∂ = ∂µ∂µ. Thisidentity is a particular case of the general identity

/A /A = γµγνAµAν =1

2(γµγν + γνγµ)AµAν = g

µνAµAν = AµAµ , (2.13)

which follows from the anticommutation relations in eqn (2.5).The condition in eqn (2.6) is necessary in order to obtain, from the Dirac

equation (2.11), a continuity equation with a quantum mechanical density

�(x) = |ψ(x)|2 = ψ†(x)ψ(x) . (2.14)To see this, we first take the Hermitian conjugate of the Dirac equation (2.11),taking into account eqn (2.8), and then multiply it on the right by γ0ψ, leading to

− i ∂µψ γ0 γµ γ0 ψ −mψψ = 0 . (2.15)Subtracting this equation from the Dirac equation (2.11) multiplied on the left byψ, we have

ψ γµ∂µ ψ + ∂µψ γ0 γµ γ0 ψ = 0 . (2.16)

Imposing the condition in eqn (2.6) we obtain the continuity equation

∂µ jµ = 0 , (2.17)

with the currentjµ(x) = ψ(x) γµ ψ(x) . (2.18)

The temporal component of this current is the quantum mechanical density ineqn (2.14). In section 2.4 it will be shown that jµ(x) is a well-behaved four-vector(see eqn (2.66) with a = b).

2.2 Representations of γ matrices

A specific choice of the four matrices γµ that satisfy the relations in eqns (2.5) and(2.6) is called a representation of the Dirac matrices. In his fundamental theorem onthe representations of the Dirac matrices, Pauli proved that all representations areunitarily equivalent, i.e. any two sets of four matrices γµ and γ′µ which fulfill therelations in eqns (2.5) and (2.6) are connected by the equivalence transformation

γ′µ = S γµ S−1 , (2.19)

where S is a unitary matrix (S† = S−1). In oder to leave the Dirac equationinvariant under a change of representation, a spinor field must transform as

ψ′ = S ψ . (2.20)

When an explicit expression of the γ matrices is needed, one can choose themost convenient one for the task under consideration. In any representation, at

10 QUANTIZED DIRAC FIELDS

most one of the γ is diagonal, because of the anticommutation relation in eqn (2.5).From eqn (2.8) it follows that the eigenvalues of γ0 are ±1 and the eigenvalues ofγk are ±i.

The standard representation of the γ matrices is the Dirac representation

γ0D =

(1 00 −1

), �γD =

(0 �σ−�σ 0

), (2.21)

in which the matrices are written as 2 × 2 blocks and we put a subscript D toindicate the Dirac representation. The 2 × 2 matrices σk are the Pauli matricesgiven in eqn (A.29).

It is useful to define the chirality matrix

γ5 ≡ γ5 ≡ i γ0 γ1 γ2 γ3 , (2.22)

which has the following useful properties:{γ5, γµ

}= 0 (2.23)(

γ5)2

= 1 (2.24)(γ5)†

= γ5. (2.25)

In the Dirac representation, the γ5 matrix is given by

γ5D =

(0 11 0

). (2.26)

For a study of relativistic particles such as neutrinos, it is convenient to usethe chiral representation in which the chirality matrix γ5C = γ

0D is diagonal and

�γC = �γD. From eqn (2.22) it follows that γ0C = −γ5D. Thus, the 4×4 Dirac matrices

in the chiral representation can be written in 2 × 2 block forms as

γ0C =

(0 −1−1 0

), �γC =

(0 �σ−�σ 0

), γ5C =

(1 00 −1

), (2.27)

or in the compact form

γµC =

(0 σ̄µ

−σµ 0

), (2.28)

with the 2 × 2 matrices

σµ = (1 , �σ) , σ̄µ = (−1 , �σ) . (2.29)

The unitary matrix SD→C which performs the equivalence transformation

γµC = SD→C γµD S−1D→C (2.30)

from the Dirac to the chiral representation is given by

SD→C =1√2

(1 + γ0D γ

5D

)=

1√2

(1 −11 1

). (2.31)

PRODUCTS OF γ MATRICES 11

2.3 Products of γ matrices

Let us define 16 matrices Γa (a = 1, 2, . . . , 16) obtained from products of γ matrices:

Γ1 ≡ 1 (no γµ matrices) , (2.32)Γ2 − Γ5 ≡ γµ (one γµ matrix) , (2.33)

Γ6 − Γ11 ≡ σµν ≡ i2

[γµ, γν ] (products of two γµ matrices) , (2.34)

Γ12 − Γ15 ≡ γµ γ5 (products of three γµ matrices) , (2.35)Γ16 ≡ γ5 (product of four γµ matrices) . (2.36)

In any product of γ matrices, all the pairs of identical matrices can be eliminatedusing the anticommutation relations in eqn (2.5). Since the number of γ matricesis four, any product of more than four γ matrices can be reduced to a product offour or fewer γ matrices.

The anticommutation relations in eqn (2.5) imply that products of γ matricesin which the matrices appear with different orders are equivalent up to a sign.Hence, the number of independent products of k γ matrices is given by the binomialcoefficient (

4k

)=

4!

k! (4 − k)! . (2.37)

A set of 16 Γ matrices represents all the irreducible products of γ matrices, asfollows

1. The matrix Γ1 = 1 is the only irreducible product with no γ matrices (( 40 ) = 1).2. The four matrices Γ2, . . . ,Γ5 = γ0, γ1, γ2, γ3 are trivially the four irreducible

products of one γ matrix (( 41 ) = 4).3. The six3 matrices Γ6, . . . ,Γ11 = σ01, σ02, σ03, σ12, σ23, σ31, represent all the

irreducible products of two γ matrices (( 42 ) = 6). In fact, we have

σµν = iγµγν (µ �= ν) . (2.38)

4. The four matrices Γ12, . . . ,Γ15 = γ0 γ5, γ1 γ5, γ2 γ5, γ3 γ5 represent all theirreducible products of three γ matrices (( 43 ) = 4), i.e.

γ0γ5 = −iγ1γ2γ3 , (2.39)γ1γ5 = −iγ0γ2γ3 , (2.40)γ2γ5 = −iγ0γ3γ1 , (2.41)γ3γ5 = −iγ0γ1γ2 , (2.42)

or, in compact form,

γαγ5 =i

3!gαβ�βµνργ

µγνγρ . (2.43)

3 By definition, the matrices σµν are antisymmetric in the indices µ, ν. Thus, thenumber of independent matrices σµν is n(n− 1)/2 = 6 for n = 4.

12 QUANTIZED DIRAC FIELDS

Table 2.1. Order of the matrices Γa and the corresponding values ofsa = sign(Γ

a)2 = 14 Tr[(Γa)2

].

a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Γa 1 γ0 γ1 γ2 γ3 σ01 σ02 σ03 σ12 σ23 σ31 γ0γ5 γ1γ5 γ2γ5 γ3γ5 γ5

sa 1 1 −1 −1 −1 −1 −1 −1 1 1 1 −1 1 1 1 1

5. The matrix Γ16 = γ5 is the only irreducible product of four γ matrices (( 44 ) = 1).

Therefore, any product of γ matrices is proportional to one of the 16 Γ matrices.The coefficient of proportionality is equal to ±1 or ±i.

The Γ matrices enjoy the following useful properties:

A. As all products of γ matrices, the product of two Γ matrices is proportional toa Γ matrix. Moreover, if two Γ matrices are different, their product is differentfrom unity:

Γa Γb ∝ Γc with Γc �= 1 for a �= b . (2.44)

If Γa and Γb are different, their product contains an odd number of at least oneγ matrix, which cannot be eliminated by using the anticommutation relationsin eqn (2.5).

B. The square of all Γ matrices is equal to ±1:

(Γa)2 = sa1 , with sa =1

4Tr[(Γa)2

]= ±1 . (2.45)

The values of sa for the 16 matrices Γa are listed in Table 2.1.

C. For each Γa with a > 1 there is at least one Γb which anticommutes with Γa,

ΓaΓb = −ΓbΓa ⇐⇒{Γa , Γb

}= 0 . (2.46)

We have

Γa = γ0 (a = 2) =⇒ Γb = γk, γ5 (b = 3, 4, 5, 16) , (2.47)Γa = γk (a = 3, 4, 5) =⇒ Γb = γ0, γ5 (b = 1, 16) , (2.48)Γa = σµν (a = 6 − 11) =⇒ Γb = σµρ ρ �= ν , (2.49)Γa = γ0γ5 (a = 12) =⇒ Γb = γkγ5 (b = 13, 14, 15) , (2.50)Γa = γkγ5 (a = 13, 14, 15) =⇒ Γb = γ0γ5 (b = 12) , (2.51)Γa = γ5 (a = 16) =⇒ Γb = γµ (b = 2, 3, 4, 5) . (2.52)

RELATIVISTIC COVARIANCE 13

D. The matrices Γa with a > 1 are traceless,

Tr[Γa] = 0 for a > 1 . (2.53)

Using a matrix Γb which anticommutes with Γa, we have

Tr[Γa] = sbTr[Γa(Γb)2]

= −sbTr[ΓbΓaΓb

]= −sbTr

[(Γb)2

Γa]

= −Tr[Γa] ,(2.54)

where the second equality has been obtained by anticommuting Γa and Γb andthe third equality has been obtained with a circular permutation of the argumentof the trace.

E. From the properties in eqns (2.44), (2.45), and (2.53) it follows that

Tr[Γa Γb

]= 4 sa δab . (2.55)

F. The Γ matrices are linearly independent4, i.e. the relation∑a

ca Γa = 0 (2.56)

implies ca = 0 for all a = 1, . . . , 16. Taking the trace of eqn (2.56) and using theproperty in eqn (2.53), one finds that c1 = 0. Similarly, taking the trace of

Γb

(∑a

ca Γa

)= 0 (2.57)

and using the property in eqn (2.55) one finds that cb = 0 for any b = 2, . . . , 16.G. From the previous properties it follows that any 4× 4 matrix X can be written

as a linear combination of the Γa matrices:

X =∑

a

xa Γa , with xa =

sa4

Tr[X Γa] . (2.58)

Therefore, the 16 matrices Γa form a basis in the vectorial space of 4×4 matrices.

2.4 Relativistic covariance

Under a Lorentz transformation in eqn (B.1), the Dirac field ψ(x) transforms as

ψ(x) → ψ′(x′) = S(Λ)ψ(x) , (2.59)with the 4 × 4 matrix S(Λ) such that

S−1(Λ) γµ S(Λ) = Λµν γν . (2.60)In this way, the Dirac equation remains form-invariant under Lorentz trans-formations, according to the principle of relativistic covariance discussed inappendix B.

4 This property implies that the minimal dimension of the Γ matrices is 4 × 4, inagreement with the minimal dimension of γ matrices discussed in footnote 2 on page 8.The reason is simply that since 4 × 4 matrices have 16 elements there are 16 independent4× 4 matrices, whereas the number of independent matrices with lower dimensionality issmaller.

14 QUANTIZED DIRAC FIELDS

The explicit form of S(1 + εω) for an infinitesimal Lorentz transformation ineqn (B.22) is

S(1 + εω) = 1 − i4ε ωµν σ

µν . (2.61)

Comparing this expression with eqn (B.63), we obtain the spin part of thegenerators of the Lorentz group in the Dirac spinor representation:

Sµν = −12σµν . (2.62)

These generators satisfy the commutation relations in eqn (B.45) of the Lorentzgroup generators.

For a finite restricted Lorentz transformation Λ = eω in eqn (B.24), we have

S(eω) = exp(− i

4ωµν σ

µν

). (2.63)

Since the adjoint field ψ(x) transform as5

ψ(x) → ψ′(x′) = ψ(x)S−1(Λ) , (2.64)

the following five Hermitian covariant bilinears transform, respectively, as a scalar,vector, antisymmetric second-rank tensor, pseudovector, and pseudoscalar: for arestricted Lorentz transformation

Sab(x) = ψa(x)ψb(x) → S′ab(x′) = ψ′a(x′)ψ′b(x′) = ψa(x)ψb(x) = Sab(x) ,(2.65)

V µab(x) = ψa(x) γµ ψb(x) → V ′µab (x′) = ψ′a(x′) γµ ψ′b(x′) = Λµν ψa(x) γν ψb(x)

= Λµν Vνab(x) , (2.66)

T µνab (x) = ψa(x)σµν ψb(x) → T ′µνab (x′) = ψ′a(x′)σµν ψ′b(x′)

= Λµα Λν

β ψa(x)σαβ ψb(x)

= Λµα Λν

β Tαβab (x) , (2.67)

Aµab(x) = ψa(x) γµ γ5 ψb(x) → A′µab(x′) = ψ′a(x′) γµ γ5 ψ′b(x′)

= Λµν ψa(x) γν γ5 ψb(x) = Λ

µν A

νab(x) ,

(2.68)

Pab(x) = ψa(x) γ5 ψb(x) → P ′ab(x′) = ψ′a(x′) γ5 ψ′b(x′) = ψa(x) γ5 ψb(x)

= Pab(x) , (2.69)

where we have considered the possibility of having different fields a and b. Thetransformations of the covariant bilinears in eqns (2.65)–(2.69) under space and

5 Indeed, we have ψ(x) = ψ†(x) γ0 → ψ′†(x) γ0 = ψ†(x)S†(Λ) γ0 = ψ(x) γ0 S†(Λ) γ0.Using the explicit form in eqn (2.61) of S(Λ) and the property in eqn (2.6), we obtain

γ0 S†(Λ) γ0 = S−1(Λ).

RELATIVISTIC COVARIANCE 15

time inversions are discussed, respectively, in sections 2.11.2 and 2.11.4. Interactionsbetween different fields (a �= b) or self-interactions (a = b) must be expressed in theLagrangian through the covariant bilinears in eqns (2.65)–(2.69), which allow us towrite scalar Lagrangian terms. The bilinears in eqns (2.65)–(2.69) are also calledcurrents.

2.4.1 Boosts

The spin part of the boost operator Kk in eqn (B.50) is

Kkspin = −1

2σ0k = − i

2γ0 γk = − i

2αk , (2.70)

where we have used the Dirac notation

αk ≡ γ0 γk , (2.71)

with {αk , αj

}= 2 δkj . (2.72)

The explicit expressions of the matrices αk in the Dirac and chiral representationsof the γ matrices are, respectively,

αkD =

(0 σk

σk 0

), αkC =

(σk 0

0 −σk). (2.73)

The matrix Skboost(ϕ) for a boost with velocity v = tanhϕ along the directionof the axis xk can be written as

Skboost(ϕ) = e−i ϕ Kkspin = e−

12 ϕ α

k

= coshϕ

2− αk sinh ϕ

2, (2.74)

where ϕ is the rapidity given in eqn (B.31), with

coshϕ

2=γ + 1

2, sinh

ϕ

2=γ − 1

2, γ =

(1 − v2

)−1/2. (2.75)

Since �α is Hermitian, Skboost(ϕ) is also Hermitian,

[Skboost(ϕ)

]†= Skboost(ϕ) , (2.76)

but not unitary. This is consistent with the fact that under a boost ψ†ψ is not ascalar, but transforms as the time-component of the four-vector ψγµψ.

16 QUANTIZED DIRAC FIELDS

2.4.2 Rotations

The spin part of the angular momentum operators Jk in eqn (B.49) is

Sk =1

4

∑j,l

�kjl σjl =1

2Σk , (2.77)

with

Σk ≡ 12

∑j,l

�kjl σjl =i

2

∑j,l

�kjl γj γl =⇒ �Σ = (iγ2γ3, iγ3γ1, iγ1γ2) .

(2.78)It is also possible to write Σk as

Σk = γ0 γk γ5 . (2.79)

The matrices satisfy the commutation and anticommutation relations

[Σk,Σj ] = 2 i∑

l

�kjl Σl , {Σk,Σj} = 2 δkj . (2.80)

The Σk matrices have the same expression in the Dirac and chiral representationsof the γ matrices:

�ΣD = �ΣC =

(�σ 00 �σ

), (2.81)

from which it is clear that the �Σ matrices represent a 4 × 4 generalization of the2 × 2 Pauli matrices.

The matrix Skrot(θ) for a rotation through an angle θ around the axis xk can bewritten as

Skrot(θ) = ei θ Sk

= ei2 θ Σ

k

= cosθ

2+ iΣk sin

θ

2. (2.82)

Since Σk is Hermitian, we have[Skrot(θ)

]†=[Skrot(θ)

]−1, (2.83)

which implies that ψ†ψ is a scalar under rotations.A rotation of 2π changes the sign of ψ, because Skrot(2π) = −1. Therefore, the

sign of ψ has no physical meaning. Indeed, all physical quantities depend on thecovariant bilinears in eqns (2.65)–(2.69), which are invariant not only under a globalchange of sign of all fermion fields, but also under a global phase transformation ofall fermion fields. This invariance corresponds, through Noether’s theorem, to theconservation of the total fermion number, which is consistent with angular momen-tum conservation: the disappearance of a fermion without creation or annihilationof another fermion would imply an unrecoverable change of 1/2 of the total angularmomentum.

HELICITY 17

2.4.3 Invariants

As discussed in section B.3 of appendix B, there are two Casimir operators ofthe Poincaré group, P 2 and W 2, which are used for the classification of itsrepresentations.

The first Casimir operator is P 2 = PµPµ, where Pµ = i∂µ is the momentum

operator (eqn (B.86)). Therefore, P 2 = −� and from the Klein–Gordon equa-tion (2.12), satisfied by the Dirac field ψ, it is clear that the eigenvalue of P 2 is m2

and the parameter m in the Dirac Lagrangian in eqn (2.1) can be interpreted asthe mass of the particle.

The second Casimir operator is W 2 = WµWµ, with the Pauli–Lubanski four-vector Wµ given in eqn (B.90), which gives the spin of the particle, according toeqn (B.95). From eqns (B.94) and (2.77), we have in the rest frame

W 0 = 0 , �W =1

2m�Σ . (2.84)

Consequently,

W 2 = −m2 14�Σ

2= −m2 3

4= −m2 1

2

(1

2+ 1

), (2.85)

which shows that a Dirac field describes particles with spin s = 1/2.

2.5 Helicity

From eqn (2.77), the helicity operator in eqn (B.96) is given by

ĥ =�S · �Ps |�P |

=�Σ · �P|�P |

. (2.86)

Since the square of the helicity operator in eqn (2.86) is one, the eigenvalues of thehelicity for a Dirac fermion are h = ±1.

As discussed at the end of section B.3 of appendix B, the mass m and the spins distinguish (possibly together with other quantum numbers) different particles.

The state of a particle is identified by the three components of the momentum �Pand by the helicity.

2.6 Gauge transformations

The Dirac Lagrangian in eqn (2.1) is invariant for global U(1) gauge transformationsof the type

ψ(x) → eiθ ψ(x) , (2.87)where θ is an arbitrary parameter. In this case, Noether’s theorem implies that theelectromagnetic current

jµ = q ψ γµ ψ (2.88)

18 QUANTIZED DIRAC FIELDS

is conserved,∂µ j

µ = 0 . (2.89)

The associated conserved charge operator is

Q =

∫d3x j0(x) = q

∫d3xψ†(x)ψ(x) , (2.90)

where q is the electric charge of the particle (see eqn (2.248)).

2.7 Chirality

The matrix γ5 is also called the chirality matrix. Since the chirality matrix isHermitian (eqn (2.25)), it can be diagonalized with a unitary transformationUγ5U † = γ5diag, with U

† = U−1. Since (γ5)2 = 1 (eqn (2.24)), the eigenvaluesof γ5 are ±1. In fact, in the chiral representation in eqn (2.27), γ5 is diagonal,equal to diag(1, 1,−1,−1).

Let us denote by ψR and ψL the fields which are eigenfunctions of γ5 with

eigenvalues +1 and −1, respectively:

γ5 ψR = + ψR , (2.91)

γ5 ψL = − ψL . (2.92)

The chiral fields ψR and ψL are called, respectively, right-handed and left-handed.It is always possible to split a generic spinor field ψ into its chiral right-handed

and left-handed components:ψ = ψR + ψL , (2.93)

with

ψR =1 + γ5

2ψ , (2.94)

ψL =1 − γ5

2ψ . (2.95)

It is convenient to define the chirality projection matrices

PR ≡1 + γ5

2, (2.96)

PL ≡1 − γ5

2, (2.97)

which satisfy the properties

PR + PL = 1 , (2.98)

(PR)2 = PR , (2.99)

(PL)2 = PL , (2.100)

CHIRALITY 19

PR PL = PL PR = 0 . (2.101)

Let us consider the Dirac Lagrangian in eqn (2.1). Using the decomposition ineqn (2.93) of the spinor field ψ, we have

L = (ψR + ψL) (i /↔∂ −m) (ψR + ψL) . (2.102)

SinceψR = PR ψ , ψL = PL ψ , (2.103)

and

P †R = PR , (2.104)

P †L = PL , (2.105)

PR γ0 = γ0 PL , (2.106)

PL γ0 = γ0 PR , (2.107)

we obtain

ψR = (PRψ) = (PRψ)†γ0 = ψ†PRγ0 = ψ†γ0PL = ψPL , (2.108)

ψL = ψPR . (2.109)

Therefore, the following four products in the Lagrangian in eqn (2.102) vanishidentically:

iψR /↔∂ψL = iψPL /

↔∂PLψ = iψ /

↔∂PRPLψ = 0 , (2.110)

iψL /↔∂ψR = 0 , (2.111)

mψRψR = mψPLPRψ = 0 , (2.112)

mψLψL = 0 . (2.113)

The Dirac Lagrangian in terms of the chiral fields ψR and ψL then becomes

L = ψR i /↔∂ ψR + ψL i /

↔∂ ψL −m

(ψR ψL + ψL ψR

). (2.114)

One can see that the chiral fields ψR and ψL have independent kinetic terms butthey are coupled by the mass term. From the Lagrangian in eqn (2.114) one canfind the field equations

i /∂ ψR = mψL , (2.115)

i /∂ ψL = mψR , (2.116)

which demonstrate that the space-time evolutions of the chiral fields ψR and ψLare related by the mass m.

The chiral fields ψR and ψL are also called Weyl spinors.

20 QUANTIZED DIRAC FIELDS

A Weyl spinor has only two independent components, as one can understandby noting that the decomposition in eqn (2.93) of a four-component spinor mustsplit the four independent components equally into two groups, one for each chiralcomponent. One can see this explicitly using a definite representation of the Diracmatrices. The most convenient one6 is the chiral representation in eqn (2.27), inwhich

PR =

(1 00 0

), PL =

(0 00 1

). (2.117)

Writing the four-component spinor ψ as

ψ =

(χRχL

), (2.118)

where χR and χL are two-component spinors, we have

ψR =

(χR0

), ψL =

(0χL

), (2.119)

showing explicitly that ψR and ψL have only two independent components.From eqns (2.28) and (2.114), the Dirac Lagrangian for the two-component fields

χR and χL is

L = iχ†R σµ∂µ χR − iχ†L σ̄µ∂µ χL +m

(χ†R χL + χ

†L χR

), (2.120)

and the field equations are

i σµ∂µ χR = −mχL , (2.121)i σ̄µ∂µ χL = mχR . (2.122)

Because

∂k =∂

∂xk= ∇k , (2.123)

these equations can be written in the explicit form

i(∂0 +�σ · �∇

)χR = −mχL , (2.124)

i(∂0 −�σ · �∇

)χL = −mχR . (2.125)

The two-component fields χR and χL are important from a relativistic point ofview because in the chiral representation the explicit expressions for the matricesσµν are

σ0kC = i

(σk 0

0 −σk), σkjC =

∑

�kj(σ 0

0 σ

). (2.126)

From eqns (2.59) and (2.63) it follows that the spinor fields χR and χL transformindependently under Lorentz transformations. This is a very important property

6 In other representations one can show that ψL has only two independent componentsusing the constraint PR ψL = 0.

CHIRALITY 21

because it implies that two-component spinors are the simplest nontrivial repre-sentations of the Lorentz group and one must consider them as the fundamentalquantities for the construction of the Lagrangian, which is a Lorentz scalar.

From the expressions in eqn (2.126) one can see that the spinor fields χR andχL transform in the same way under rotations, discussed in section 2.4.2. For arotation through an angle θ around the axis xk we have

χR,L →(

cosθ

2+ i σk sin

θ

2

)χR,L . (2.127)

On the other hand, the spinor fields χR and χL transform in different ways underLorentz boosts, discussed in section 2.4.1. For a boost with velocity v = tanhϕalong the direction of the axis xk we have

χR →(cosh

ϕ

2− σk sinh ϕ

2

)χR , (2.128)

χL →(cosh

ϕ

2+ σk sinh

ϕ

2

)χL . (2.129)

Therefore, the spinor fields χR and χL belong to two different representations of theLorentz group, which are traditionally called dotted and undotted (see, for example,Ref. [866]).

2.7.1 Massless field

From eqns (2.115) and (2.116) one can see that the space-time evolutions of thechiral fields ψR and ψL decouple for m = 0. In this case, we obtain the Weylequations

i /∂ ψR = 0 , (2.130)

i /∂ ψL = 0 . (2.131)

Since the field equations of the chiral fields ψR and ψL are decoupled, the chiralfields are independent and it is possible that one of the two chiral fields is sufficientfor the description of a massless fermion.

Let us consider a solution ψ(x, p) of the massless Dirac equation

i /∂ ψ(x, p) = 0 , (2.132)

which is also an eigenfunction of the four-momentum operator Pµ = i∂µ,

Pµ ψ(x, p) = pµ ψ(x, p) , (2.133)

with energyp0 = E = |�p| . (2.134)

In this case, the massless Dirac equation (2.132) can be written as(γ0 |�p| −�γ ·�p

)ψ(x, p) = 0 . (2.135)

22 QUANTIZED DIRAC FIELDS

Multiplying this equation on the left by γ5γ0 and using eqn (2.79), we obtain

�Σ ·�p|�p| ψ(x, p) = γ

5 ψ(x, p) . (2.136)

Now we can see that the operator on the left-hand side is nothing but the helicityoperator in eqn (2.86), with Pµ → pµ. Equation (2.136) shows that chirality coin-cides with helicity for the eigenfunctions of the four-momentum which are solutionsof the massless Dirac equation. In particular, the eigenfunctions of the chiralitymatrix are eigenfunctions of the helicity operator with the same eigenvalue:

�Σ ·�p|�p| ψR(x, p) = ψR(x, p) , (2.137)

�Σ ·�p|�p| ψL(x, p) = −ψL(x, p) . (2.138)

Hence, a massless right-handed chiral field ψR(x, p) with a definite four-momentumhas positive helicity, and a massless left-handed chiral field ψL(x, p) with a definitefour-momentum has negative helicity.

2.8 Solution of the Dirac equation

The Dirac equation (2.11) can be solved using the Fourier expansion of the Diracfield

ψ(x) =

∫d3p

(2π)3 2E

∑h=±1

[a(h)(p)u(h)(p) e−ip ·x + b(h)†(p) v(h)(p) eip ·x

], (2.139)

where h is the helicity and

p0 = E =

√�p2 +m2 , (2.140)

in order to satisfy the Klein–Gordon equation (2.12). The quantities u(h)(p), v(h)(p)are spinors and a(h)(p), b(h)(p) are numerical coefficients. We have adopted thenotation

a(h)†(p) ≡ a(h)∗(p) and b(h)†(p) ≡ b(h)∗(p) (2.141)for later convenience in the discussion of the quantized Dirac field (in section 2.9).

The phase-space measure in the Fourier expansion of ψ(x) in eqn (2.139) isinvariant under restricted Lorentz transformations. This can be seen by writing itas

d3p

(2π)3 2E=

d4p

(2π)42π δ(p2 −m2) θ(p0) , (2.142)

where we have used the property in eqn (A.124) of the Dirac δ-function, whichimplies

δ(p2 −m2) = δ(p02 − E2) = δ(p0 − E) + δ(p0 + E)

2E. (2.143)

Note that restricted Lorentz transformations do not change the sign of p0.

SOLUTION OF THE DIRAC EQUATION 23

From the Dirac equation in eqn (2.11) one can see that the spinors u(h)(p) andv(h)(p) satisfy the equations

(/p−m)u(h)(p) = 0 , (2.144)

(/p+m) v(h)(p) = 0 . (2.145)

For the adjoint spinors, using the property in eqn (2.6), we obtain

u(h)(p) (/p−m) = 0 , (2.146)

v(h)(p) (/p+m) = 0 . (2.147)

From eqns (2.144)–(2.147) it follows that

u(h)(p) v(h′)(p) = 0 . (2.148)

Let us now derive the helicity properties of u(h)(p) and v(h)(p). The field ψ(x)can be written as the sum

ψ(x) =∑

h=±1ψ(h)(x) , (2.149)

where

ψ(h)(x) =

∫d3p

(2π)3 2E

[a(h)(p)u(h)(p) e−ip ·x + b(h)†(p) v(h)(p) eip ·x

](2.150)

is an eigenfield of the helicity operator in eqn (2.86) with eigenvalue h:

ĥ ψ(h)(x) = hψ(h)(x) . (2.151)

Applying the helicity operator in eqn (2.86) to ψ(h)(x) we find

ĥ ψ(h)(x) =�Σ · �P|�P |

ψ(h)(x)

=

∫d3p

(2π)3 2E

[a(h)(p)

�Σ · �p|�p| u

(h)(p) e−ip ·x − b(h)†(p)�Σ · �p|�p| v

(h)(p) eip · x].

(2.152)

In order to satisfy the eigenvalue equation (2.151), u(h)(p) and v(h)(p) must be

eigenfunctions of the helicity operator in momentum space �Σ · �p/|�p| with oppositeeigenvalues:

�p ·�Σ|�p| u

(h)(p) = hu(h)(p) , (2.153)

�p ·�Σ|�p| v

(h)(p) = −h v(h)(p) . (2.154)

24 QUANTIZED DIRAC FIELDS

The corresponding equations for the adjoint spinors are, with eqn (2.6),

u(h)(p)�p ·�Σ|�p| = hu

(h)(p) , (2.155)

v(h)(p)�p · �Σ|�p| = −h v

(h)(p) . (2.156)

From eqns (2.153)–(2.156) it follows that u(h)(p)u(h′)(p) ∝ δhh′ and

v(h)(p)v(h′)(p) ∝ δhh′ . Here, we adopt the Lorentz-invariant normalization

conditions

u(h)(p)u(h′)(p) = 2mδhh′ , (2.157)

v(h)(p) v(h′)(p) = −2mδhh′ . (2.158)

From these normalization conditions and the properties in eqns (2.144)–(2.147),one can derive the useful relations7

u(h)(p) γµ u(h′)(p) = v(h)(p) γµ v(h

′)(p) = 2 pµ δhh′ , (2.159)

u(h)(p) γ5 u(h′)(p) = v(h)(p) γ5 v(h

′)(p) = 0 , (2.160)

u(h)†(p) v(h′)(pP) = v

(h)†(p)u(h′)(pP) = 0 . (2.161)

where pP = (p0,−�p).

Using the relations in eqns (2.159) and (2.161), one can find that the coefficientsa(h)(p) and b(h)(p) in eqn (2.139) are given by

a(h)(p) =

∫d3xu(h)†(p)ψ(x) eip ·x , (2.162)

b(h)(p) =

∫d3xψ†(x) v(h)(p) eip · x . (2.163)

The normalization ∫d3x |ψ(x)|2 = 1 (2.164)

implies that the coefficients a(h)(p) and b(h)(p) are constrained by∫d3p

(2π)3 2E

∑h=±1

[|a(h)(p)|2 + |b(h)(p)|2

]= 1 . (2.165)

7 For example,

u(h)(p) γµ u(h′)(p) = u(h)(p)

γµ/p+ /pγµ

2mu(h

′)(p) =pµ

mu(h)(p)u(h

′)(p) = 2 pµ δhh′ ,

and

u(h)(p) γ5 u(h′)(p) = u(h)(p) γ5

/p

mu(h

′)(p) = −u(h)(p)/p

mγ5 u(h

′)(p)

= − u(h)(p) γ5 u(h′)(p) = 0 .

SOLUTION OF THE DIRAC EQUATION 25

Since the four spinors u(+)(p), u(−)(p), v(+)(p), v(−)(p) are mutually orthog-onal, they are linearly independent and form a basis of the vector space offour-dimensional spinors. The outer products

u(+)(p)u(+)(p) , u(−)(p)u(−)(p) , v(+)(p) v(+)(p) , v(−)(p) v(−)(p) (2.166)

form a basis of the vector space of 4 × 4 matrices. In particular, they satisfy thecompleteness relation

∑h=±1

[u(h)(p)u(h)(p)

2m− v

(h)(p) v(h)(p)

2m

]= 1 . (2.167)

The components of ψ(x) proportional to e−ip · x and eip · x are usually called,respectively, positive-energy and negative-energy components because (P 0 = i∂0)

P 0 e−ip ·x = E e−ip · x , P 0 eip ·x = −E eip ·x . (2.168)

It is useful to define the projection operators on the components with positive andnegative energy:

Λ±(p) ≡m± /p2m

, (2.169)

with ∑r=±

Λr(p) = 1 , Λr(p) Λs(p) = Λr(p) δrs , (2.170)

and

Λ+(p)u(h)(p) = u(h)(p) , Λ+(p) v

(h)(p) = 0 , (2.171)

Λ−(p)u(h)(p) = 0 , Λ−(p) v(h)(p) = v(h)(p) . (2.172)

From these equations and the completeness relation in eqn (2.167) one can derivethe useful identities8

Λ+(p) =∑

h=±1

u(h)(p)u(h)(p)

2m, (2.173)

Λ−(p) = −∑

h=±1

v(h)(p) v(h)(p)

2m. (2.174)

From eqns (2.155) and (2.156), the projection operators on the u and v spinorswith definite helicity are, respectively,

P(u)h =

1

2

(1 + h

�p · �Σ|�p|

), (2.175)

8 For example, multiplying Λ+(p)u(h)(p) on the right by u(h)(p), summing over h and

expressingP

h u(h)(p)u(h)(p) on the left-hand side as 2m+

Ph v

(h)(p)v(h)(p), one obtainsthe identity in eqn (2.173).

26 QUANTIZED DIRAC FIELDS

P(v)h =

1

2

(1 − h�p ·

�Σ

|�p|

). (2.176)

It is possible to write these projection operators in a unified covariant form. Byusing the property in eqn (2.144) and eqns (2.79), (A.64), we have

�p · �Σ|�p| u

(h)(p) =�p · �Σ|�p|

/p

mu(h)(p) =

γ5 γ0�γ ·�p|�p|

E γ0 −�γ ·�pm

u(h)(p)

= γ5( |�p|mγ0 − E

m

�γ ·�p|�p|

)u(h)(p) = h γ5 /sh u

(h)(p) , (2.177)

where sµh is the polarization four-vector

sµh = h

( |�p|m,E

m

�p

|�p|

), (2.178)

with

s2h = −1 , sh · p = 0 . (2.179)

In a similar way, one can obtain

�p ·�Σ|�p| v

(h)(p) = −h γ5 /sh v(h)(p) . (2.180)

Hence, the helicity projection operators in eqn (2.175) and (2.176) can be writtenin a unified covariant form

Ph =1 + γ5 /sh

2. (2.181)

The orthogonality of sh and p guarantees that Ph commutes with Λ± because[γ5 /sh , /p

]= γ5 {/sh , /p} −

{γ5 , /p

}/sh = 0 . (2.182)

Therefore, we can define the four projection operators on the components withdefinite energy and helicity as

Λh±(p) ≡ Λ±(p)Ph = Ph Λ±(p) =(m± /p2m

)(1 + γ5 /sh

2

), (2.183)

such that ∑r=±

∑h=±1

Λhr (p) = 1 , Λhr (p) Λ

h′

s (p) = Λhr (p) δrs δhh′ , (2.184)

and

Λh+(p)u(h′)(p) = δhh′ u

(h′)(p) , Λh+(p) v(h′)(p) = 0 , (2.185)

Λh−(p)u(h′)(p) = 0 , Λh−(p) v

(h′)(p) = δhh′ v(h′)(p) . (2.186)

SOLUTION OF THE DIRAC EQUATION 27

From these equations and the completeness relation in eqn (2.167) one can derivethe useful identities (using a method similar to that explained in footnote 8 onpage 25)

Λh+(p) =u(h)(p)u(h)(p)

2m, (2.187)

Λh−(p) = −v(h)(p) v(h)(p)

2m. (2.188)

Let us finally derive some useful relations among the u and v spinors withdifferent helicities. Since

(/pP −m) γ0 u(−h)(pP) = 0 and�p · �Σ|�p| γ

0 u(−h)(pP) = h γ0 u(−h)(pP) ,

(2.189)we have

γ0 u(−h)(pP) = η(�p, h)u(h)(p) , (2.190)

where η(�p, h) is a phase factor which depends on �p and h. Changing the signsof �p and h we obtain γ0 u(h)(p) = η(−�p,−h)u(−h)(pP), which is compatible witheqn (2.190) only if

η(−�p,−h) = η∗(�p, h) . (2.191)

A relation similar to eqn (2.190) holds for the v’s. In the treatment of chargeconjugation, to be discussed in section 2.11.1, it is convenient to choose the relativephase of the spinors u(h)(p) and v(h)(p) in order to satisfy the relation in eqn (2.354).In this case, we obtain

γ0 v(−h)(pP) = −η∗(�p, h) v(h)(p) . (2.192)

Furthermore, since

(/p−m) γ5 v(−h)(p) = 0 and �p ·�Σ

|�p| γ5 v(−h)(p) = h γ5 v(−h)(p) , (2.193)

we have

γ5 v(−h)(p) = ζ(h)u(h)(p) , (2.194)

where ζ(h) is a phase factor which depends on h. A similar relation for γ5 u(−h)(p)is constrained by the relation in eqn (2.354) to be

γ5 u(−h)(p) = −ζ∗(h) v(h)(p) . (2.195)

This is compatible with eqn (2.194) only if

ζ(−h) = −ζ(h) . (2.196)

28 QUANTIZED DIRAC FIELDS

2.8.1 Dirac representation

The explicit form of the free spinors u(h)(p) and v(h)(p) in the Dirac representationof the γ matrices (see eqn (2.21)) is

u(h)D (p) =

( √E +mχ(h)(�p)

h√E −mχ(h)(�p)

), (2.197)

v(h)D (p) =

(−√E −mχ(−h)(�p)

h√E +mχ(−h)(�p)

), (2.198)

where χ(h)(�p) are the orthonormal two-component helicity eigenstate spinors to bediscussed in section 2.8.3. One can easily find that ζ(h) in eqns (2.194) and (2.195)is given by

ζ(h) = −h . (2.199)

In the nonrelativistic limit, |�p| � m, the spinors in eqns (2.197) and (2.198) areapproximated by

u(h)D (p) �

√2m

(χ(h)(�p)

h |p|2m χ(h)(�p)

), (2.200)

v(h)D (p) =

√2m

(− |p|2m χ(−h)(�p)hχ(−h)(�p)

). (2.201)

Here, the two upper components of u(h)(p), called large components, are much largerthan the two lower components, called small components. The opposite is true forv(h)(p). The Dirac representation is convenient to study nonrelativistic problemsbecause of these properties.

2.8.2 Chiral representation

The explicit form of the free spinors u(h)(p) and v(h)(p) in the chiral representationof the γ matrices (see eqn (2.27)) is

u(h)C (p) =

(−√E + h |�p|χ(h)(�p)√E − h |�p|χ(h)(�p)

), (2.202)

v(h)C (p) = − h

(√E − h |�p|χ(−h)(�p)√E + h |�p|χ(−h)(�p)

), (2.203)

where χ(h)(�p) are the orthonormal two-component helicity eigenstate spinors dis-cussed in section 2.8.3. One can see that ζ(h) in eqns (2.194) and (2.195) has the

SOLUTION OF THE DIRAC EQUATION 29

same expression in eqn (2.199) as in the Dirac representation,

ζ(h) = −h . (2.204)

In the relativistic limit m� E the spinors in eqns (2.202) and (2.203) become

u(+)C (p) � −

√2E

(χ(+)(�p)

− m2E χ(+)(�p)

), u

(−)C (p) �

√2E

(− m2E χ(−)(�p)χ(−)(�p)

),

(2.205)

v(+)C (p) � −

√2E

(m2E χ

(−)(�p)χ(−)(�p)

), v

(−)C (p) �

√2E

(χ(+)(�p)

m2E χ

(+)(�p)

), (2.206)

with two of the four components suppressed by the small ratio m/E. For this reasonthe chiral representation is convenient in the treatment of ultrarelativistic particles.

2.8.3 Two-component helicity eigenstate spinors

In sections 2.8.1 and 2.8.2 χ(h)(�p) are two-component helicity eigenstate spinorsthat satisfy the eigenvalue equation

�p ·�σ|�p| χ

(h)(�p) = hχ(h)(�p) . (2.207)

Hence, u(h)(p) and v(h)(p) in eqns (2.197), (2.198) and (2.202), (2.203) satisfythe eigenvalue equations (2.153) and (2.154). The eigenvalue equation (2.207)guarantees that χ(h)(�p) and χ(−h)(�p) are orthogonal,(

χ(h)(�p))†χ(−h)(�p) = 0 . (2.208)

They are also chosen to be normalized to one:(χ(h)(�p)

)†χ(h)(�p) = 1 . (2.209)

In the treatment of charge conjugation, to be discussed in section 2.11.1, it isconvenient to choose the relative phase of χ(h)(�p) and χ(−h)(�p) in order to satisfythe relation in eqn (2.354). In this case, in both Dirac and chiral representationswe obtain the relation

iσ2(χ(h)(�p)

)∗= −hχ(−h)(�p) . (2.210)

Since χ(−h)(−�p) is an eigenfunction of �p ·�σ/|�p| with eigenvalue h, it is proportionalto χ(h)(�p):

χ(−h)(−�p) = η(�p, h)χ(h)(�p) , (2.211)where η(�p, h) is the same phase factor as in eqn (2.190). Changing the sign of both�pand h in eqn (2.211) we obtain χ(h)(�p) = η(−�p,−h)χ(−h)(−�p), which is compatible

30 QUANTIZED DIRAC FIELDS

with eqn (2.211) only if

η(−�p,−h) = η∗(�p, h) , (2.212)

in agreement with eqn (2.191). Furthermore, the compatibility of eqns (2.211) and(2.210) gives the constraint

η(�p,−h) = −η∗(�p, h) . (2.213)

Let us finally remark that the two-component helicity eigenstate spinors satisfy theuseful relation9 (

χ(h)(�p))†

σk χ(h)(�p) = hpk

|�p| . (2.214)

Using polar coordinates θ and φ, with 0 ≤ θ ≤ π and 0 ≤ φ < 2π, the three-momentum is written as �p = |�p| (sinθ cosφ, sinθ sinφ, cosθ) and the two-componenthelicity eigenstate spinors are given by

χ(+)(�p) =

(cos θ2

sin θ2 eiφ

), χ(−)(�p) =

(− sin θ2 e−iφ

cos θ2

). (2.215)

Since the transformation �p → −�p is equivalent to a transformation θ → π − θ andφ → φ ± π, with the plus sign if 0 ≤ φ < π and the minus sign if π ≤ φ < 2π, wehave

η(�p, h) = h e−ihφ . (2.216)

It is often convenient to orient�p along the z axis: �p = (0, 0, |�p|), which implies θ = 0and

χ(+)(�p) =

(10

)≡ χ(+) , χ(−)(�p) =

(01

)≡ χ(−) . (2.217)

This is also a convenient form for the two-component spinors in the rest frame ofthe particle, where the helicity is undetermined. In the following, we will adoptsuch a definition.

2.8.4 Massless field

From eqns (2.205) and (2.206) it is clear that in the case of a massless fermion thefour spinors u(±)(p) and v(±)(p) have only two nonvanishing components. In thissubsection we discuss the implications of this property for the massless chiral fieldsψR and ψL.

9 From eqns (2.207) and (A.35) we have

“χ(h)(�p)

”†σkχ(h)(�p) =

“χ(h)(�p)

”† (�p ·�σ)σk + σk(�p ·�σ)2h|�p|

χ(h)(�p)

=pj

2h|�p|

“χ(h)(�p)

”†{σk, σj}χ(h)(�p) =

pk

h|�p|.

QUANTIZATION 31

The Fourier expansions of the chiral fields ψR and ψL are given by

ψR,L(x) =

∫d3p

(2π)3 2E

∑h=±1

[a(h)(p)u

(h)R,L(p) e

−ip · x + b(h)†(p) v(h)R,L(p) eip ·x].

(2.218)From eqns (2.205) and (2.206), for m = 0 we have

u(+)L (p) = u

(−)R (p) = v

(+)R (p) = v

(−)L (p) = 0 , (2.219)

and

u(+)R (p) = u

(+)(p) , u(−)L (p) = u

(−)(p) , v(+)L (p) = v(+)(p) , v

(−)R (p) = v

(−)(p) .(2.220)

Therefore, the Fourier expansions of the massless chiral fields ψR and ψL simplifyto

ψR(x) =

∫d3p

(2π)3 2E

[a(+)(p)u(+)(p) e−ip ·x + b(−)†(p) v(−)(p) eip ·x

], (2.221)

ψL(x) =

∫d3p

(2π)3 2E

[a(−)(p)u(−)(p) e−ip ·x + b(+)†(p) v(+)(p) eip ·x

]. (2.222)

One can see explicitly that the massless chiral fields ψR(x) and ψL(x) are indepen-dent, in agreement with the discussion in section 2.7.1. Moreover, the positiveenergy components of ψR(x) and ψL(x) have positive and negative helicity,respectively, in agreement with eqns (2.137) and (2.138).

2.9 Quantization

The quantization of the Dirac field can be implemented by imposing the canonicalequal-time anticommutation relations in section C.2 of appendix C. Special care isnecessary for the determination of the canonical conjugated momentum, becausethe Dirac Lagrangian in eqn (2.1) contains ∂0ψ and ∂0ψ

†, which are not indepen-dent. Separating the real and imaginary parts of each component ψα of ψ, andtaking into account that eπα = ∂L /∂(∂0eψα) and �mπα = −∂L /∂(∂0�mψα),one can find that

π = i ψ† . (2.223)

Thus, the anticommutation relations in eqns (C.11) and (C.12) can be written as

{ψα(t,�x), ψ†β(t,�y)} = δ3(�x−�y) δαβ , (2.224)

{ψα(t,�x), ψβ(t,�y)} = {ψ†α(t,�x), ψ†β(t,�y)} = 0 , (2.225)

where α, β are Dirac indices. The anticommutation relations in eqn (2.224) and(2.225) are satisfied by writing the Fourier expansion of the quantized Dirac