Elementary particle physics: vol 1 quantum field theory and particles

Yorikiyo NagashimaElementary Particle Physics Volume 1: Quantum Field Theory and Particles WILEY-VCH Verlag GmbH & Co... Elementary Particle Physics Volume 2: Standard Model and Experime

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Yorikiyo Nagashima

Elementary Particle Physics

Volume 1: Quantum Field Theory and Particles

WILEY-VCH Verlag GmbH & Co KGaA

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Related Titles

Nagashima, Y

Volume 2: Standard Model and Experiments

Approx 2013

ISBN 978-3-527-40966-2

Russenschuck, S

Field Computation for Accelerator Magnets

Analytical and Numerical Methods for Electromagnetic Design and Optimization

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Volume 1: Quantum Field Theory and Particles

WILEY-VCH Verlag GmbH & Co KGaA

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Thus particle physics covers a vast area To master it, you usually have to readdifferent books, at increasingly advanced levels, on individual subjects like its his-torical background, basic experimental data, quantum ﬁeld theory, mathematicalconcepts, theoretical models, cosmological concepts, etc This book covers most ofthose topics in a single volume in an integrated manner Not only that, it showsyou how to derive each important mathematical formula in minute detail, thenasks you to work out many problems, with answers given at the end of the book.Some abstract formulas are immediately followed by their intuitive interpretationand experimental consequences The same topics are often repeated at differentlevels of sophistication in different chapters as you read on, which will help deepenyour understanding.

All these features are quite unique to this book, and will be most helpful to dents as well as laymen or non-experts who want to learn the subject seriously andenjoy it It can serve both as a text book and as a compendium on particle physics.Even for practicing particle physicists and professors this will be a valuable refer-ence book to keep at hand Few people like Professor Nagashima, an accomplishedexperimental physicist who is also conversant with sophisticated theoretical sub-jects, could have written it

Elementary Particle Physics, Volume 1: Quantum Field Theory and Particles Yorikiyo Nagashima

ISBN: 978-3-527-40962-4

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1.1 An Overview of the Standard Model 3

1.1.1 What is an Elementary Particle? 3

1.1.2 The Four Fundamental Forces and Their Uniﬁcation 4

1.1.3 The Standard Model 7

1.2 The Accelerator as a Microscope 11

2 Particles and Fields 13

2.2.4 Intuitive Picture of a Field and Its Quantum 28

2.2.5 Mechanical Model of a Classical Field 29

3.2.2 Lorentz Vectors and Scalars 43

3.3 Space Inversion and Time Reversal 45

3.4 Covariant Formalism 47

3.4.1 Tensors 47

3.4.2 Covariance 48

3.4.3 Supplementing the Time Component 49

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4.1.3 Interpretation of the Negative Energy 64

4.1.4 Lorentz-Covariant Dirac Equation 69

4.2 Plane-Wave Solution 71

4.3 Properties of the Dirac Particle 75

4.3.1 Magnetic Moment of the Electron 75

5.2.1 Heisenberg Equation of Motion 100

5.2.2 Quantization of the Harmonic Oscillator 102

6.2 Asymptotic Field Condition 131

6.3 Explicit Form of the S-Matrix 133

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Contents IX

6.5.1 Transition Rate 141

6.5.2 Relativistic Normalization 142

6.5.3 Incoming Flux and Final State Density 144

6.5.4 Lorentz-Invariant Phase Space 145

6.5.5 Cross Section in the Center of Mass Frame 145

6.6 Vertex Functions and the Feynman Propagator 147

8 Radiative Corrections and Tests of Qed* 191

8.1 Radiative Corrections and Renormalization* 191

8.1.1 Vertex Correction 191

8.1.2 Ultraviolet Divergence 193

8.1.3 Infrared Divergence 197

8.1.4 Infrared Compensation to All Orders* 199

8.1.5 Running Coupling Constant 204

9.1.1 Space and Time Translation 223

9.1.2 Rotational Invariance in the Two-Body System 227

9.2 Discrete Symmetries 233

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9.3.4 SU(2) (Isospin) Symmetry 260

10 Path Integral: Basics 267

10.3 Feynman’s Path Integral 274

10.3.1 Sum over History 274

10.3.2 Equivalence with the Schrödinger Equation 278 10.3.3 Functional Calculus 279

10.4 Propagators: Simple Examples 282

10.6.2 Note on Imaginary Time 302

10.6.3 Correlation Functions as Functional Derivatives 304

10.7 Connection with Statistical Mechanics 306

11 Path Integral Approach to Field Theory 311

11.1 From Particles to Fields 311

11.2 Real Scalar Field 312

11.3.1 Gauge Fixing and the Photon Propagator 321

11.3.2 Generating Functional of the Electromagnetic Field 323

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12.4 Particle Interactions with Matter 378

12.4.1 Some Basic Concepts 378

12.4.2 Ionization Loss 381

12.4.3 Multiple Scattering 389

12.4.4 Cherenkov and Transition Radiation 390

12.4.5 Interactions of Electrons and Photons with Matter 394

12.4.6 Hadronic Shower 401

12.5 Particle Detectors 403

12.5.1 Overview of Radioisotope Detectors 403

12.5.2 Detectors that Use Light 404

12.5.3 Detectors that Use Electric Signals 410

12.5.4 Functional Usage of Detectors 415

12.6 Collider Detectors 422

12.7 Statistics and Errors 428

12.7.1 Basics of Statistics 428

12.7.2 Maximum Likelihood and Goodness of Fit 433

12.7.3 Least Squares Method 438

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13.6 Low-Energy Nuclear Force 487

13.6.1 Spin–Isospin Exchange Force 487

14.1.1 The Discovery of Strange Particles 502

14.1.2 The Sakata Model 505

14.5.5 Lifetime of Charmed Particles 536

14.5.6 Charm Spectroscopy: SU(4) 537

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Contents XIII

14.5.7 The Fifth Quark b (Bottom) 539

14.6 Color Charge 539

14.6.1 Color Independence 542

14.6.2 Color Exchange Force 544

14.6.3 Spin Exchange Force 545

14.6.4 Mass Formulae of Hadrons 547

15.3 Chirality of the Leptons 567

15.3.1 Helicity and Angular Correlation 567

15.3.2 Electron Helicity 569

15.4 The Neutrino 571

15.4.1 Detection of the Neutrino 571

15.4.2 Mass of the Neutrino 572

15.4.3 Helicity of the Electron Neutrino 576

15.4.4 The Second Neutrino ν μ 578

15.5 The Universal V–A Interaction 579

15.7.4 The Generation Puzzle 605

15.8 A Step Toward a Uniﬁed Theory 608

15.8.1 Organizing the Weak Phenomena 608

15.8.2 Limitations of the Fermi Theory 610

15.8.3 Introduction of SU(2) 614

16 Neutral Kaons and CP Violation* 617

16.1 Introduction 618

16.1.1 Strangeness Eigenstates and CP Eigenstates 618

16.1.2 Schrödinger Equation for K0K0States 619

16.1.3 Strangeness Oscillation 622

16.1.4 Regeneration of K1 626

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16.4 Test of T and CPT Invariance 653

16.4.1 Deﬁnition of T- and CPT-Violating Amplitudes 654 16.4.2 T Violation 654

17.3 e–p Elastic Scattering 683

17.4 Electron Proton Deep Inelastic Scattering 687

17.4.1 Cross-Section Formula for Inelastic Scattering 687 17.4.2 Bjorken Scaling 690

17.5 Parton Model 693

17.5.1 Impulse Approximation 693

17.5.2 Electron–Parton Scattering 696

17.6 Scattering with Equivalent Photons 699

17.6.1 Transverse and Longitudinal Photons 699

17.6.2 Spin of the Target 702

17.6.3 Photon Flux 703

17.7 How to Do Neutrino Experiments 705

17.7.1 Neutrino Beams 705

17.7.2 Neutrino Detectors 709

17.8 ν –p Deep Inelastic Scattering 712

17.8.1 Cross Sections and Structure Functions 712 17.8.2 ν, ν–q Scattering 715

17.8.3 Valence Quarks and Sea Quarks 716

17.8.4 Comparisons with Experimental Data 717

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18.2.3 Parallel Transport and Connection 734

18.2.4 Rotation in Internal Space 737

18.6 Uniﬁed Theory of the Electroweak Interaction 770

18.6.1 S U(2) U(1) Gauge Theory 770

18.6.2 Spontaneous Symmetry Breaking 774

18.6.3 Higgs Mechanism 778

18.6.4 Glashow–Weinberg–Salam Electroweak Theory 782

18.6.5 Summary of GWS Theory 784

19 Epilogue 787

19.1 Completing the Picture 788

19.2 Beyond the Standard Model 789

Appendix A Spinor Representation 803

A.1 Deﬁnition of a Group 803

A.1.1 Lie Group 804

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XVI Contents

Appendix B Coulomb Gauge 813

B.1 Quantization of the Electromagnetic Field in the Coulomb Gauge 814

Appendix C Dirac Matrix and Gamma Matrix Traces 817

C.1 Dirac Plane Wave Solutions 817

C.2 Dirac γ Matrices 817

C.2.1 Traces of the γ Matrices 818

C.2.2 Levi-Civita Antisymmetric Tensor 819

C.3 Spin Sum of jMf ij2 819

C.3.1 A Frequently Used Example 820

C.3.2 Polarization Sum of the Vector Particle 822

C.4 Other Useful Formulae 823

Appendix D Dimensional Regularization 825

D.1 Photon Self-Energy 825

D.2 Electron Self-Energy 830

Appendix E Rotation Matrix 833

E.1 Angular Momentum Operators 833

E.2 Addition of the Angular Momentum 835

E.3 Rotational Matrix 835

Appendix F C, P, T Transformation 839

Appendix G S U(3), S U(n) and the Quark Model 841

G.1 Generators of the Group 841

Appendix H Mass Matrix and Decaying States 859

H.1 The Decay Formalism 859

Appendix I Answers to the Problems 865

Appendix J Particle Data 915

Appendix K Constants 917

References 919

Index 929

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Preface

The purpose of this book is to present introductory explanations to junior ate students of major advances in particle physics made in the last century It isalso aimed at laypeople who have not received a standard education in physics butare willing to make an extra effort to follow mathematical logic Therefore it is or-ganized to be suitable for self-study and is as self-contained as possible This iswhy it starts with an introduction to field theory while the main purpose is to talkabout particle physics Quantum field theory is an essential mathematical tool forunderstanding particle physics, which has fused quantum mechanics with specialrelativity However, phenomena that need the theory of relativity are almost com-pletely limited to particle physics, cosmology and parts of astrophysics, which arenot topics of everyday life Few students dare to take a course in special or gen-eral relativity This fact, together with the mathematical complexity of field theory,makes particle physics hard to approach If one tries to make a self-contained expla-nation of mathematical aspects of the field, hardly any space is left for the details ofparticle-physics phenomena While good textbooks on field theories are abundant,very few talk about the rich phenomena in the world of elementary particle physics.One reason may be that the experimental progress is so rapid that any textbook thattries to take an in-depth approach to the experimental aspect of the field becomesobsolete by the time it is published In addition, when one makes a great effort tofind a good book, basic formulae are usually given without explanation The au-thor had long felt the need for a good textbook suitable for a graduate course, wellbalanced between theory and experiment

gradu-Without doubt many people have been fascinated by the world that quantummechanics opens up After a student has learned the secrets of its mystery, he(or she) must realize that his world view has changed for life But at what depthwould it have been if he thought he had grasped the idea without knowledge ofthe Schödinger equation? This is why the author feels uneasy with presentation ofthe basic formulae as given from heaven, as is often the case in many textbooks onexperimental particle physics

This book is the ﬁrst of two volumes Volume I paves the way to the StandardModel and Volume II develops applications and discusses recent progress

Volume I, i.e this book, is further divided into two parts, a ﬁeld theoretical proach and the way to the Standard Model The aim of Part I is to equip students

ap-Elementary Particle Physics, Volume 1: Quantum Field Theory and Particles Yorikiyo Nagashima

ISBN: 978-3-527-40962-4

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XVIII Preface

with tools so they can calculate basic formulae to analyze the experimental datausing quantized ﬁeld theories at least at the tree level Tailored to the above pur-pose, the author’s intention is to present a readable introduction and, hopefully, anintermediate step to the study of advanced ﬁeld theories

Chapters 1 and 2 are an introduction and outline of what the Standard Model is.Chapter 3 picks out essential ingredients of special relativity relevant for particlephysics and prepares for easy understanding of relativistic formulae that follow.Chapters 4–7, starting from the Dirac equation, ﬁeld quantization, the scatteringmatrix and leading to QED (quantum electrodynamics), should be treated as one,closed subject to provide basic knowledge of relativistic ﬁeld theories and step-by-step acquisition of the necessary tools for calculating various dynamic processes.For the treatment of higher order radiative corrections, only a brief explanation isgiven in Chapter 8, to discuss how mathematical consistency of the whole theory

is achieved by renormalization

Chapter 9 deals with space-time and internal symmetry, somewhat independenttopics that play a double role as an introduction to the subject and appendices toparticle physics, which is discussed in Part II

Two chapters, 10 and 11, on the path integral are included as an addition Themethod is very powerful if one wants to go into the formal aspects of ﬁeld theory,including non-Abelian gauge theories, in any detail Besides it is the modern way

of interpreting quantum mechanics that has many applications in various ﬁelds,and the author felt acquaintance with the path integral is indispensable However,from a practical point of view of calculating the tree-level formulae, the traditionalcanonical quantization method is all one needs With all the new concepts andmathematical preparation, the chapter stops at a point where it has rederived whathad already been calculated before So readers have the choice of skipping thesechapters and coming back later when their interest has been aroused

By jumping to Chapter 18 which describes axioms of the Standard Model, theﬁeld theoretical approach can be concluded in a self-contained way, at least formal-

ly But it was placed at the end of Part II because the phenomenological approachalso culminates in the formulation of the uniﬁed gauge theories Although two al-ternative approaches are prepared to reach the goal, the purpose of the book is topresent particle physics, and ﬁeld theory is to be considered as a tool and treated assuch

As part of the minimum necessary knowledge, a chapter was added at the end ofPart I on basic techniques of experimental measurements, the other necessary tool

to understand particle physics phenomena Readers who are perplexed with thecombination may skip this chapter, but it is the author’s wish that even studentswho aim at a theoretical career should understand at least this level of experimentaltechniques After all, the essence of natural science is to explain facts and theoristsshould understand what the data really mean In addition, a few representativeexperiments are picked out and scattered throughout the book with appropriatedescriptions They were chosen for their importance in physics but also in order todemonstrate how modern experiments are carried out

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Chapter 16 on CP violation in K mesons is an exception to the outline describedabove It was included for two reasons First, because it is an old topic, dating back

to 1964, and second, because it is also a new topic, which in author’s personal ception may play a decisive role in the future course of particle physics However,discussions from the viewpoint of uniﬁed theories is deferred until we discuss therole of B-physics in Volume II Important though it is, its study is a side road fromthe main route of reaching the Standard Model, and this chapter should be con-sidered as a topic independent of the rest; one can study it when one’s interest isaroused

per-Although the experimental data are all up-to-date, new phenomena that the dard Model predicted and were discovered after its establishment in the early 1970sare deferred to Volume II The last chapter concludes what remains to be done onthe Standard Model and introduces some intriguing ideas about the future of par-ticle physics

Stan-Volume II, which has yet to come, starts with the axioms of gauge ﬁeld ries and presents major experimental facts that the Standard Model predicted andwhose foundations were subsequently established These include W, Z, QCD jetphenomena and CP violation in B physics in both electron and hadron colliders.The second half of Volume II deals issues beyond the Standard Model and recentdevelopments of particle physics, the Higgs, the neutrino, grand uniﬁed theories,unsolved problems in the Standard Model, such as axions, and the connection withcosmology

theo-A knowledge of quantum mechanics and the basics of special relativity is quired to read this book, but that is all that is required One thing the author aimed

re-at in writing this series is to make a reference book so thre-at it can serve as a sort

of encyclopedia later on For that reason, each chapter is organized to be as contained as possible The list of references is extensive on purpose Also, as a re-sult, some chapters include sections that are at higher level than that the reader haslearned up to that point An asterisk * is attached to those sections and problems

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Acknowledgements

The author wishes to thank Professor J Arafune (Tokyo University) for valuableadvice and Professors T Kubota and T Yamanaka (Osaka University) for readingpart of the manuscript and for giving many useful suggestions Needless to say, anymistakes are entirely the author’s responsibility and he appreciates it if the readernotiﬁes them to him whenever possible

The author would like to express his gratitude to the authors cited in the textand to the following publishers for permission to reproduce various photographs,ﬁgures and diagrams:

American Institute of Physics, publisher of Physics Today for permission to produce Figure 19.4;

re- American Physical Society, publisher of the Physical Review, Physical ReviewLetters and the Review of Modern Physics, for permission to reproduce Figures4.3, 5.2ab, 8.8, 9.2, 9.3ab, 12.18abc, 13.11a, 13.17, 13.19, 13.21, 13.23, 13.24ab,14.6, 14.13–15, 14.20b, 14.25, 14.30, 15.3ab, 15.13, 15.15, 15.17a, 15.21, 16.5–7ab, 16.19, 16.21–23, 17.2b–17.7, 17.12c, 17.15a, 17.21ab, 18.10ab, 18.11abc;

Annual Reviews, publisher of Annual Review of Nuclear and Particle Sciencefor permission to reproduce Figures 12.23ab, 13.29, 14.18, 17.11;

Elsevier Science Ltd., publisher of Nuclear Physics, Physics Letters, Physics port, for permission to reproduce Figures 8.9ab, 12.24, 12.25, 12.29, 12.33ab,12.35ab, 12.37, 12.40–42, 13.14abcd, 13.18, 13.30, 15.8, 15.17b, 15.22, 15.23,16.1b, 16.8ab–16.10ab, 16.12, 16.13, 16.15–17ab, 17.19, 18.18;

Re- Institute of Physics Publishing Ltd., publisher of Report on Progress of Physicsfor permission to reproduce Figures 15.11 and 15.12;

Japan Physical Society, publisher of Butsuri for permission to reproduce Figure12.26b;

Nature Publishing Group, publisher of Nature for permission to reproduce ures 13.5 and 14.1;

Fig- Particle Data Group, publisher of Review of Particle Physics for permission to produce Figures 12.10, 12.15–12.17, 12.20, 12.28b, 12.30ab, 12.38, 12.47, 13.25,13.26, 14.11, 14.16, 14.24, 14.26, 17.17, 18.16;

re- Royal Society, publisher of the Proceedings of the Royal Society for permission

to reproduce Figure 13.4;

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XXII Acknowledgements

Shokabo, publisher of the Cosmic Rays for permission to reproduce Figure12.22;

Springer Verlag, publisher of Zeitschrift für Physik and European Journal

of Physics for permission to reproduce Figures 12.43, 16.18, 17.14c, 17.15b,17.18ab, 17.24

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Part One A Field Theoretical Approach

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What is an Elementary Particle?

Particle physics, also known as high-energy physics, is the ﬁeld of natural sciencethat pursues the ultimate structure of matter This is possible in two ways One is tolook for elementary particles, the ultimate constituents of matter at their smallestscale, and the other is to clarify what interactions are acting among them to con-struct matter as we see them The exploitable size of microscopic objects becomessmaller as technology develops What was regarded as an elementary particle atone time is recognized as a structured object and relinquishes the title of “elemen-tary particle” to more fundamental particles in the next era This process has beenrepeated many times throughout the history of science

In the 19th century, when modern atomic theory was established, the exploitablesize of the microscopic object was 1010m and the atom was “the elementaryparticle” Then it was recognized as a structured object when J.J Thomson extract-

ed electrons in 1897 from matter in the form of cathode rays Its real structure (the

Rutherford model) was clariﬁed by investigating the scattering pattern of

α-parti-cles striking a golden foil In 1932, Chadwick discovered that the nucleus, the core

of the atom, consisted of protons and neutrons In the same year, Lawrence structed the ﬁrst cyclotron In 1934 Fermi proposed a theory of weak interactions

con-In 1935 Yukawa proposed the meson theory to explain the nuclear force actingamong them

It is probably fair to say that the modern history of elementary particles beganaround this time The protons and neutrons together with their companion pions,which are collectively called hadrons, were considered as elementary particles un-til ca 1960 We now know that they are composed of more fundamental particles,

the quarks Electrons remain elementary to this day Muons and τ-leptons, which

were found later, are nothing but heavy electrons, as far as the present

technolo-gy can tell, and they are collectively dubbed leptons Quarks and leptons are thefundamental building blocks of matter The microscopic size that can be explored

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4 1 Introduction

by modern technology is nearing 1019m The quarks and leptons are elementary

at this level They may or may not be at a smaller level The popular string theoryregards the most fundamental matter constituent not as a particle but as a string

at the Planck scale ( 1035m), but in this book we limit ourselves to treating onlyexperimentally established facts and foreseeable extensions of their models

1.1.2

The Four Fundamental Forces and Their Unification

Long-Range Forces: Gravity and the Electromagnetic Force There are four kinds offundamental forces in nature, the strong, electromagnetic, weak and gravitationalforces The strong force, as the name suggests, is the strongest of all; it is a thou-sand times stronger, and the weak force a thousand times weaker, than the electro-magnetic force The gravitational force or gravity is negligibly weak (that betweenthe proton and electron is 1042 times the electromagnetic force) at elementaryparticle levels Its strength becomes comparable to the others only on an extreme-

ly small scale (Planck scale 1035m), or equivalently at extremely high energy(Planck energy 1019GeV), not realizable on the terrestrial world Despite its in-trinsic weakness, gravity controls the macroscopic world Indeed, galaxies hundredmillions of light years apart are known to be mutually attracted by the gravita-tional force The strength of the force decreases (only) inversely proportional tothe distance squared, but it also scales as the mass, and the concentration of themass more than compensates the distance even on the cosmic scale We refer to it

as a long-range force The behavior of the electromagnetic force is similar to andfar stronger than gravity, but, in general, the positive and negative charges com-pensate each other However, its long-distance effect manifests itself in the form

of the galactic magnetic ﬁeld, which spans hundreds or possibly millions of lightyears It binds nuclei and electrons to form atoms, atoms to form molecules andmolecules to form matter as we observe them Namely, it is the decisive force inthe microworld where the properties of matter are concerned Atomic, molecu-lar and condensed matter physics need only consider the electromagnetic force Amathematical framework, called quantum electrodynamics (QED), is used to de-scribe the dynamical behavior of point-like charged particles, notably electrons, theelectromagnetic ﬁeld and their interactions Mathematically, it is a combination ofquantized Maxwell equations and relativistic quantum mechanics

Short-Range Forces: The Strong and Weak Forces In the ultramicroscopic world ofthe nuclei, hadrons and quarks (at scales less than 1015m), both the strong andthe weak force come in The reason why they are important only at such a smallscale is that they are short ranged reaching only a few hadron diameters, and ac-cumulation of the mass does not help to make it stronger They are referred to asshort-range forces The strong force (referred to as the color force at the most fun-damental level) acts between quarks to bind them to form hadrons Historically, thestrong force was ﬁrst discovered as the nuclear force to bind protons and neutrons

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1.1 An Overview of the Standard Model 5

nized as a kind of molecular force (van der Waals force) that can be derived fromthe more fundamental color force In 1935, Yukawa predicted the existence of the

πmeson as the carrier of the nuclear force The idea that the force is transmitted

by a “force carrier particle” was revolutionary and laid the foundation for day gauge theories Later, it was clariﬁed that the pion was a composite and cannot

present-be a fundamental force carrier, but the basic idea remains valid The weak force is

known to act in the decay of hadrons, notably in nuclear β decays It is also known

to control the burning rate of the sun and to play a decisive role to in the explosion

of type II supernovae

The electromagnetic force, though playing an essential role at the microscopiclevel, can also act at the macroscopic distance We observe electromagnetic phe-nomena in daily life, for example electrically driven motors or the reaction of a tinymagnet to the earth’s magnetic ﬁeld The reason is that its strength is proportion-

al to the inverse square of the distance (Coulomb’s law) In general, a long-range

force decreases only as the inverse power of the distance ( r n) and can act on anobject no matter how far away it is if there is enough of the force source A short-range force, on the other hand, is a force whose strength diminishes exponentially

with the distance The nuclear force, for instance, behaves like e r/r0 /r2(r0

1015m) and reaches only as far as r r0 The range of the weak force is evenshorter, of the order of 1018m Because of this, despite their far greater strengththan gravity, their effects are not observed in the macroscopic world However, ifone can conﬁne enough matter within the reach of the strong force and let it re-act, it is possible to produce a large amount of energy Nuclear power and nuclearfusion are examples of such applications

When a Long-Range Force Becomes a Short-Range Force The above description phasizes the difference between the properties of the forces Modern uniﬁed theo-ries tell us, however, that all the fundamental forces, despite their variety of appear-ances, are essentially long-range forces and can be treated within the mathematicalframework of gauge theories The seemingly different behavior is a result of meta-morphosis caused by environmental conditions We introduce here one example of

em-a long-rem-ange force becoming short rem-ange under certem-ain circumstem-ances A trem-ain with

a linear induction motor, the next generation supertrain, uses a phenomenon calledthe Meissner effect to magnetically levitate its body in the air There, the magnet-

ic field is completely repelled from the superconductor This happens because, atvery low temperatures near absolute zero, electron pairs in a special configurationcalled the “Cooper pair” collectively react to the magnetic field to form a strong eddycurrent to cancel the invading magnetic flux The cancellation is complete exceptfor a shallow depth at the surface, into which the flux can invade In other words,the electromagnetic force, which is originally long range, becomes short range inthe superconductor This example inspired Nambu [287] to formulate an importantconcept known as “spontaneous symmetry breaking”, which played a crucial role

in forming the Standard Model

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6 1 Introduction

Higgs Mechanism The essence of modern uniﬁed theory is to recognize that “thevacuum in which we live is in a sort of superconducting phase” What corresponds

to the Cooper pair in the superconductor is called the Higgs particle, unseen by

us but thought to permeate everywhere in the universe The weak force is longrange originally, but because of the ultracold vacuum that surrounds us becomesshort range, owing to the pseudo-Meissner effect caused by the Higgs particles.The short-range force translates to a ﬁnite mass of the force carrier particle inquantum ﬁeld theory, as we shall see in the following sections The Higgs parti-cle condenses at the ultracold temperature and causes a phase transition of thevacuum, a phenomenon similar to the vapor-to-water phase transition Elementaryparticles, which have zero mass originally, acquire mass because of the draggingeffect caused by swimming through the condensed sea of the Higgs particle Thismass-generating scheme is called the Higgs mechanism It is thought that at hightemperature (such as at the hot Big Bang) and before the Higgs condensation theweak force was a long-range force, just like the electromagnetic force, and bothwere just different aspects of the same force The relation between them is similar

to that between magnetic and electric forces They are uniﬁed in the sense theyare described by a single mathematical framework, just like the electric and mag-netic forces are described by the Maxwell equations After Glashow, Weinberg andSalam, who constructed the uniﬁed theory of the electromagnetic and the weakforces, it is often called as GWS theory or simply the electroweak theory

Confinement and Screening of Color Contrary to the weak force, the strong forcebecomes short range for a different reason A group of quarks bind together bythe color force to become hadrons Protons and pions are examples At short dis-tances, the color force between the quarks is inversely proportional to the square

of the distance, just like any long-distance forces but at long distances, its behavior

is different and has a component that does not diminish as the distance increases.Therefore, stored potential energy between them increases with distance, hence

an inﬁnite energy is required to separate them to a macroscopic distance Thequarks are, in a sense, connected by a string and, like the north and south poles

of a magnet, cannot be isolated individually, a phenomenon called “conﬁnement”.They tend not to separate from each other by more than 1015m, which is thetypical size of hadrons When the string is stretched beyond its limit, it is cut intomany pieces, producing more hadrons The color charges of the quarks come inthree kinds dubbed “R, G, B”, for red, green and blue, but only a neutral color com-bination can form hadrons.1)

In the vicinity of the hadron, quarks of different colorare located at different distances and hence the effect of individual color can be felt.But from far away the color looks completely neutralised and the strong force be-tween the hadrons vanishes, making it a short-range force The force between thehadrons, including the nuclear force, is an action of partially compensated residual1) The name “color” is used as a metaphor for three kinds of force-generating charge We may equally call them by a number or any other index Only a certain mixture of R, G, B, or color with anticolor (complementary color) that makes white can form hadrons Hence in the color terminology we can conveniently say hadrons are color-neutral.

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colors and is therefore secondary in its nature, like the van der Waals force actingbetween molecules The action of color forces is, mathematically, almost identical

to that of the electromagnetic force and is also described by a gauge theory, thedifference being only in the number of charges (three).2)

This is the reason for thename quantum chromodynamics (QCD) to describe the mathematical framework

of the strong force dynamics

1.1.3

The Standard Model

The weak force can be described by a gauge theory that contains two kinds ofcharge General relativity, the theory of gravity, was also recognized as a kind ofgauge theory Considering that the electromagnetic and weak forces are unifiedand all four fundamental forces work in the same mathematical framework, it isnatural to consider that all the forces are unified but show different aspects in dif-ferent environments The Grand Unified Theory (GUT), to unify the electroweakand strong interactions, and the Super Grand Unified Theory,3)

to combine all theforces, including gravity, are currently active research areas Among them “super-gravity” and “superstring theories” are the most popular But only the electroweaktheory and QCD are experimentally well established and are called collectively “theStandard Model” (SM) of elementary particles The history and current situation offorce uniﬁcation are shown in Fig 1.1

The essence of the Standard Model can be summarized as follows:

1 The elementary particles of matter are quarks and leptons

2 The mathematical framework for the force dynamics are gauge theories

3 The vacuum is in a sort of superconducting phase

Particle species in the Standard Model are given in Table 2.1 of Sect 2.1 Matter ticles, the quarks and leptons, are fermions having spin 1/2, and the fundamentalforce carriers, also called gauge particles, are spin 1 bosons The Higgs boson is aspin 0 particle and is the only undiscovered member Whether it is elementary or

par-a dynpar-amicpar-al object rempar-ains to be seen The Stpar-andpar-ard Model wpar-as proposed par-around

1970 and was ﬁrmly established experimentally by the end of the 1970s The dard Model was constructed on the torrent of data from the 1960s to 1980s, whenlarge accelerators become operational (see Table 1.1) It marked the end of the era

Stan-of the 80-year quest for the fundamental building blocks Stan-of matter that started withRutherford at the beginning of the 20th century, and opened a new epoch Theeffect on our consciousness (material view) was revolutionary and comparable tothat of relativity and quantum mechanics at the beginning of the 20th century The2) Gluons, the equivalent of photons in electromagnetism, come in eight colors, which are

combinations of the three basic colors, excluding white.

3) The preﬁx “super” is attached for theories that contain the supersymmetry that combines

fermions and bosons to form one family They are examples of theories beyond the Standard Model and are brieﬂy described in Chap 19.

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8 1 Introduction

Strong Force Electro-magnetic Weak Force Gravity (Nuclear Force) Force

Strength ~0.1 (~10) 1/137 ~10 ~10 # -5 -42 Source Color Charge Electric Charge Weak Charge Mass

Classical Newton's Law Theory Maxwell Equation

General Relativity

1935 1934

Yukawa Theory Fermi Theory 1946~1949

Quantum Quantum Electro-Dynamics

Theory (Quark Model) (QED)

Quantum Chromo-

Dynamics (QCD) Salam Theory

Standard Model

Grand Unified Theory ? New Kaluza-Klein Theory

Super Grand Unified Theory String Theory ?

Unification of the Forces

~1973 1960~1968

1665 1864

1914

# At r = 10 m -18

Figure 1.1 Unification of the forces Those enclosed in dashed lines are not yet established.

Only the strong force changes its strength appreciably within currently available energy ranges

or, equivalently, distances This is why the distance of the strong force is specified.

Standard Model is very powerful and can explain all the phenomena in the croworld of particles in a simple and uniﬁed way, at least in principle Forty yearshave passed since the Standard Model was established, yet there has appeared on-

mi-ly one phenomenon that goes beyond the Standard Model Neutrino oscillation inwhich a neutrino of one kind is transformed to another while it propagates, doesnot happen if the mass of the neutrino vanishes, as is assumed in the StandardModel But it is fair to say it only needs a small stretch and is not a contradiction

to the Standard Model The model is so powerful to the extent that it is not easy

to think of an experiment with currently available accelerators that could challengethe Standard Model in a serious way A quantum jump in the accelerator energyand/or intensity is required to ﬁnd phenomena that go beyond the Standard Modeland explore new physics

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Table 1.1 Chronicle of particle physics Important discoveries and contributing accelerators.

Items in parentheses ( ) are theoretical works.

Year Names Discovered or proposed Accelerator

1929 W Heisenberg (Quantum ﬁeld theory)

W Pauli

1930 P.A.M Dirac (Dirac equation)

1934 E Fermi (Weak interaction theory)

1935 H Yukawa (Meson theory)

S.H Neddermyer

" G.O Rochester Strange particles Cosmic ray

C.C Butler

1946–1949 S Tomonaga (QED)

J Schwinger

R Feynman

1950–1955 Strange particle production Cosmotron

1953 K Nishijima (Nishijima–Gell-Mann law)

by bubble chambers Cosmotron

M Schwarz

J Steinberger

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10 1 Introduction

Table 1.1 (continued)

Year Names Discovered or proposed Accelerator

1964 M Gell-Mann (Quark model)

G Zweig

J Cronin

" P Higgs (Higgs mechanism)

1961–1968 S Glashow (Electro-weak uniﬁcation)

" A Lagarrigue et al Neutral current CERN/PS

van de Meer

Precision test of EW theory LEP, TEVATRON

1998 Y Totsuka et al Neutrino oscillation Cosmic rays

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1.2 The Accelerator as a Microscope 11

The fact that the kinetic energy of particles in particle physics far exceeds themass energy makes special relativity an indispensable tool to describe their behav-ior Understanding the mathematical formalism that is built on the combination ofspecial relativity and quantum mechanics, also known as quantum ﬁeld theory, re-quires a fair amount of effort, and the ensuing results very often transcend humancommon sense The only other ﬁelds that need both disciplines are cosmology andextreme phenomena in astrophysics, such as supernovae and black-hole dynam-ics

1.2

The Accelerator as a Microscope

In order to investigate the microscopic structure of matter, we need a tool, a scope, that magnifies the object so that we can see it, or more generally recognize itusing our five senses What corresponds to the light-collecting device in the micro-scope is a particle accelerator Shining light on the object corresponds to makinghigh-energy particles collide with target particles whose structure we want to inves-tigate The scattered light is collected by objective lenses and focused to an image,which we observe through an eyepiece In the accelerator experiment, we use aradioisotope detector to measure patterns of the scattered particles By computerprocessing they can be transformed to a pattern that we can recognize as a mag-nified image The magnification of a microscope is determined by a combination

micro-of lenses, but they merely transform the image to help human eyes to recognizethe scattered pattern of light The resolution, the ability of the optical microscope

to discriminate, is largely determined by the wavelength of the light The morepowerful electron microscope uses an electron beam and instead of optical lenses,quadrupole magnets to focus the beam The scattered electrons are illuminated on

a ﬂuorescent board to make an image that human eyes can recognize In otherwords, the electron microscope is a mini accelerator–detector complex

The quantum principle dictates that particles have a dual wave/particle propertyand the wavelength is given by the Planck constant divided by the particle momen-

tum (the de Broglie wavelength λ D h/p ) Therefore, the resolving power is directly

proportional to the momentum or the energy of the incoming particle This is why

we need high-energy accelerators to explore the structure of matter The smaller thescale, the higher the energy Therefore, to explore elementary particles, the highestenergy technologically realizable at the era is required This is why particle physics

is often called high-energy physics

Figure 1.2, commonly referred to as the Livingston chart, shows the history ofmajor accelerators and their energy On the right are indicated the approximateranges of energy that are suitable for the investigation of the particle structure atthe speciﬁed level We summarize important discoveries and tools used for them inthe history of particle physics in Table 1.1 We can see that the most effective toolsfor particle physics are accelerators of the highest energy available at the time Theone with the highest energy currently is the LHC (Large Hadron Collider; a proton–

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12 1 Introduction

Figure 1.2 The Livingstone chart showing development of accelerators and the investigated

constituent particles.

proton colliding accelerator of total energy 14 TeV) in Europe It started operation

in 2009 The LHC can explore a micro-size of 1020m and is expected to discoverthe Higgs particle

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2

Particles and Fields

In particle physics, a particle is deﬁned as an energy quantum associated with a

“field” The field is an entity that is defined over all space and time (collectivelycalled space-time) and is able to produce waves, according to classical physics, orquanta, in present day terminology, when excited or when some amount of energy

is injected In a microworld the quanta are observed as particles, but collectivelythey behave like waves A quantum is the name given to an object that possessesthe dual properties of both a particle and a wave A typical example of a field is theelectromagnetic field It is created by a charge and extends all over space It is staticwhen the charge that creates it does not move, but it can be excited by vibratingthe source of the field, the electric charge; then the vibrating field propagates This

is a charge radiating an electromagnetic wave It is well known that historicallyquantum mechanics has its origin in Planck’s recognition that blackbody radiation

is a collection of countable quanta, photons

In 19th century physics, waves could only be transmitted by some kind of brating medium and the electromagnetic waves were considered to propagate in amedium called the “ether” The existence of the ether was ruled out with the advent

vi-of special relativity, and people began to consider that the vacuum, though void vi-ofanything in the classical sense, has the built-in property of producing all kinds offields, whose excitations are observed as quanta or particles The Standard Modelhas advanced the idea that the vacuum is not an empty entity but rather filled withvarious kinds of exotic material, including the Higgs particle, and exhibits dynam-ical properties like those observed in ordinary matter The vacuum as we view itnowadays is a kind of resurrected ether with strange attributes that nobody hadthought of In this chapter, we describe an intuitive picture of both particles andfields, define some associated variables and prepare basic tools and terminologiesnecessary for the treatment of quantum field theory

2.1

What is a Particle?

There are several properties that characterize a particle

ISBN: 978-3-527-40962-4

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14 2 Particles and Fields

(1) A particle is spatially localized at each instant and its number is countable.

In classical mechanics, particle’s mass m and velocity v or energy E and mentum p can be deﬁned and measured In the nonrelativistic limit, where the

mo-particle’s velocity is far smaller than the velocity of light c, these variables of a free

moving particle are given by

2m v

2D p2

where bold letters denote three-component vectors However, when the velocity is

close to c, Einstein’s formulae

E2D(p c)2C(m c2)2, v D p c

2

Note, Eq (2.3) also holds for particles with zero mass, such as the photon When

the mass vanishes, E D p c and v is always equal to c.

(2) The particle can be created or annihilated.

This results from Eq (2.2) The mass of a particle is but one form of energy andcan be created if enough energy exists, or the energy can be converted back to massenergy subject to selection rules of one kind or another Examples of reactions are

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con-2.1 What is a Particle? 15

(3) A particle is not necessarily stable.

The statement is a corollary to statement (2) The Equation (2.4e) says that the

pion is unstable and decays to a muon and a neutrino Its average lifetime τ is

measured to be approximately 2.6 108s This means there is a time uncertainty

Δt τ for the existence of the pion Then the Heisenberg uncertainty principle

tells us that there is also uncertainty in the energy or the mass in the rest frame

Here „ D h/2π is Planck’s constant divided by 2π and is also called Planck’s

constant when there is no confusion In the case of the pion

Δmc2ΔE  „

Δt

6.58 1022MeV s2.6 108s 2.5 10

Let us treat the situation a little more quantitatively When the average lifetime

is given by τ, the particle can die in a brief time interval d t with a probability d t/τ.

If d N of N particles die in d t

d N

d t τ

) N(t) D N0e t/τ

(2.9)

In quantum mechanics, the existence probability of a particle is proportional to the

square of its wave function ψ

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16 2 Particles and Fields

the time dependence of the probability is given by

jψ(t)j2D jψ(x)e i(E0i Γ /2)t/„j2D jψ(x)j2e Γ t/„

τ

(2.12)

From the equations, we know that being unstable is equivalent to having a complex

energy with negative imaginary part Γ > 0 The next question is then what kind

of mass distribution does the particle have? The wave function as a function of theenergy can be expressed as a Fourier transform

Figure 2.1 Lorentzian shape or Breit–Wigner resonance formula for the energy (or mass)

distri-bution of unstable particles.

Tiêu đề	Elementary Particle Physics Volume 1: Quantum Field Theory and Particles
Tác giả	Yorikiyo Nagashima
Trường học	Wiley-VCH Verlag GmbH & Co. KGaA
Thể loại	book
Năm xuất bản	Approx. 2013
Thành phố	Wiley-VCH

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Số trang	966
Dung lượng	8,87 MB