Quarks, partons and Quantum Chromodynamics Lecture notes on

Quarks,partons

and

Quantum Chromodynamics

Lecture notes on the phenomenology of the quark-parton model and Quantum

Chromodynamics1

Jiˇr′?Ch′y la

Institute of Physics,Academy of Sciences of the Czech Republic

Abstract

The elements of the quark-parton model and its incorporation within the framework of perturbative Quantum Chromodynamics are laid out.The development of the basic concept of quarks as fundamental constituents of matter is traced in considerable detail,emphasizing the interplay of experimental observations and their theoretical interpretation.The relation of quarks to partons,invented by Feynman to explain striking phenomena in deep inelastic scattering of electrons on nucleons,is explained.This is followed by the introduction into the phenomenology of the parton model,which provides the basic means of communication between experimentalists and theorists.

No systematic exposition of the formalism of quantized?elds is given,but the essential di?erences of Quantum Chromodynamics and the more familiar Quantum Electrodynamics are pointed out.The ideas of asymptotic freedom and quark con?nement are introduced and the central role played in their derivation by the color degree of freedom are analyzed.

Considerable attention is paid to the physical interpretation of ultraviolet and infrared singularities in QCD,which lead to crucial concepts of the“running”coupling constant and the jet,respectively.Basic equations of QCD–improved quark–parton model are derived and the main features of their solutions dis-cussed.It is argued that the“pictures”taken by modern detectors provide a compelling evidence for the interpretation of jets as traces of underlying quark–gluon dynamics.Finally,the role of jets as tools in searching for new phenomena is illustrated on several examples of recent important discoveries.

1Available at http://www-hep.fzu.cz/chyla/lectures/text.pdf

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Contents

1Introduction7

1.1What are these lectures all about? (7)

1.2Rutherford experiment (8)

1.3Bohr model of atom (10)

1.4Scattering experiment and the concept of cross-section (10)

1.5Scattering in nonrelativistic classical physics (12)

1.6Scattering in nonrelativistic quantum physics (14)

1.7Scattering on bound systems (15)

1.8Exercises (16)

2Elements of groups and algebras17

2.1De?nitions and basic facts (17)

2.2Lie groups and algebras (19)

2.3SU(2)group and algebra (21)

2.4Application of SU(2)group:the isospin (23)

2.5Weights and roots of compact Lie algebras (25)

2.6Simple roots of simple Lie algebras (28)

2.7Examples of further important multiplets of SU(3) (32)

2.7.1Sextet (33)

2.7.2Octet (33)

2.7.3Decuplet (34)

2.7.4Other products and multiplets (34)

2.8Exercises (34)

3The road to quarks36

3.1What are the nuclei made of,what forces hold them together? (36)

3.1.1Electrons within nuclei (36)

3.1.2First glimpse of nuclear force (37)

3.1.3Birth of the“new”quantum mechanics (38)

3.1.4Pauli and neutrino hypothesis (38)

3.1.51932-34:three years that changed all (40)

3.1.61935-1938:?nal touches (42)

3.21938-1947:decade of uncertainty (42)

3.31947-1955:strange discovery and its consequences (43)

3.41956-1960:paving the eightfold way (47)

3.51961-1964:from resonances to quarks (50)

4Quark Model54

4.1January1964:birth of the quark model (54)

4.2Quarks with?avor and spin:the SU(6)symmetry (56)

4.3Spin structure of the baryons (61)

4.4The Zweig rule (62)

4.5Problems and puzzles of the Quark Model (62)

4.6Color to the rescue:Quasinuclear colored model of hadrons (64)

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4.7The arrival of charm (66)

4.8To be or not to be? (71)

4.9Lone at the top (73)

4.10Exercises (74)

5Elements of the parton model75

5.1Kinematics (75)

5.2Dynamics (76)

5.3*Cross–sections of virtual photons (81)

5.4Electrons as tools for investigating nuclear structure (83)

5.5Probing the structure of proton by elastic ep scattering (84)

5.6What does the deep inelastic scattering of electrons tell us about the structure of nucleons?.85

5.7Emergence of the parton model (89)

5.8Parton distribution functions and their basic properties (93)

5.9Parton model in neutrino interactions (96)

5.10Polarized nucleon structure functions (100)

5.11Space–time picture of DIS and hadronization (102)

5.12Deep inelastic scattering at small x (105)

5.13Exercises (105)

6Parton model in other processes107

6.1Electron-positron annihilations into hadrons at high energies (107)

6.2Drell-Yan production of heavy dilepton pairs (110)

6.3Exercises (112)

7Basics of Quantum Chromodynamics113

7.1The rise and fall of quantum?eld theory (113)

7.2And its remarkable resurrection (115)

7.3Maxwell equations in the covariant form (117)

7.4Comments on the basic idea and concepts of gauge theories (118)

7.5Elementary calculations (124)

7.5.1Quark-quark scattering (124)

7.5.2Quark-gluon scattering (125)

7.5.3Gluon-gluon scattering (126)

7.5.4Comparison of di?erent parton subprocesses (127)

7.6Exercises (128)

8Ultraviolet renormalization–basic ideas and techniques129

8.1*In?nities in perturbative calculations (129)

8.2*Renormalization in QED (130)

8.2.1Electric charge renormalization (130)

8.2.2*Photon wave function renormalization (138)

8.2.3*Renormalization of the electron mass (139)

8.2.4*Vertex renormalization (142)

8.3Elements of dimensional regularization (143)

8.3.1A simple example (143)

8.3.2Feynman parametrization (147)

8.3.3Momentum integrals and related formulae (147)

8.3.4Dimensional regularization in QED and QCD (148)

8.3.5*Renormalization at one loop level and the technique of counterterms (150)

8.4*Renormalization in QCD (151)

8.4.1*Higher orders,IR?xed points and consistency of perturbation theory (154)

8.4.2Quark mass thresholds in the runningαs (156)

8.4.3*Summation of QCD perturbation expansions (158)

8.5Exercises (163)

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9Mass singularities and jets165

9.1*Mass singularities in perturbation theory (165)

9.2*Kinoshita–Lee–Nauenberg theorem (168)

9.3*Introduction into the theory of jets (170)

9.3.1Jet algorithms (171)

9.3.2Jets of partons vs jets of hadrons (172)

9.4Exercises (173)

10QCD improved quark-parton model174

10.1General framework (174)

10.2Partons within partons (176)

10.2.1Equivalent photon approximation (176)

10.2.2Branching functions in QCD (179)

10.2.3*Multigluon emission and Sudakov formfactors of partons (180)

10.3Partons within hadrons (182)

10.3.1Evolution equations at the leading order (182)

10.3.2Moments of structure functions and sum rules (185)

10.3.3Extraction of the gluon distribution function from scaling violations (186)

10.3.4*Evolution equations at the next-to-leading order (187)

10.3.5*Hard scattering and the factorization theorem (189)

10.4Brief survey of methods of solving the evolution equations (190)

10.5Exercises (192)

11Particle interactions as seen in modern detectors194

11.1The H1experiment at DESY (194)

11.1.1Manifestation of the jet–like structure of?nal states in DIS (196)

11.1.2Photoproduction of jets (198)

11.1.3Photoproduction of the J/ψparticle (199)

11.1.4Are all jets alike? (200)

11.2Top quark discovery at Fermilab (200)

11.3Are quarks composite? (200)

11.4W pair production at LEP (204)

12Appendix205

12.1Dirac equation and traces ofγmatrices (205)

12.2SU c(3)matrices and color traces (206)

12.3Feynman diagram rules in QCD (207)

12.4Special functions (208)

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Prolog

This text is an extended version of the lectures read in a one-semester course to students at the Faculty of Mathematics and Physics of the Charles University in Prague.Some parts of it go,however,beyond the introductory level and could be used by postgraduate students.These more advanced parts concern primarily questions related to various aspects of the renormalization procedure.

The reader is expected to be familiar with the technique of Feynman diagrams,but no special ability in carrying out actual calculations is assumed.On many places I emphasize the intuitive interpretation of these diagrams in terms of the “virtual particles”(or,more appropriately,“virtual states”).I am convinced that this interpretation helps in deriving various formulae and approximations and provides deeper physical insight into the meaning of perturbation theory.

Throughout the text the system of units conventional in elementary particle physics is used.In these units =c =1,masses,momenta and energies are measured in GeV and the lengths are given in GeV ?1.The conversion factor reads:1fm=10?13cm .=5GeV ?1.

There is a number of books that should be consulted for more in-depth exposition of formal aspects of the elements of quantum ?eld theory.Out of the textbooks on foundations of quantum ?eld theory the recently published three–volume

S.Weinberg:The Quantum Theory of Fields [1]

by Steven Weinberg has rapidly become a modern classic.Beside formal aspects of quantum ?eld theory it contains in the second volume numerous modern applications as well.Despite its age and,consequently,absence of any reference to quarks and partons,I still recommend any beginner to look into the good old

J.D.Bjorken &S.D.Drell:Relativistic Quantum Theory [2]

for a very physical introduction to the Feynman diagram technique.To get a feeling of the power of the modern language of path integrals in their derivation I recommend a small,but marvelous book

R.Feynman:QED,strange theory of light and matter [3].

The basics of the quark-parton model can be found in many books,but the original text

R.Feynman:Photon-hadron Interactions [4],

as well as another classic book

F.Close:An Introduction to Quarks and Partons [5],

are still highly recommended.The chapter on group theory is based on an excellent book

H.Georgi:Lie Algebras in Particle Physics [6],

tailored speci?cally to the needs of an active particle physicist.To get a proper historical perspective of the developments of subatomic physics up to early ?fties the reader should consult the book

S.Weinberg:Discovery of Subatomic Particles [7],

written in the same spirit as the well known First three minutes by the same author.Detailed and instructive description of the crucial experiments on deep inelastic scattering of electrons on nucleons,carried out at SLAC in late sixties,can be found in 1992Nobel lectures of J.Friedman,W.Kendall and R.Taylor [8].

There is a number of other good books or lecture notes covering parts of this course,among them

Ta-Pei Cheng,Ling-Fong Li:Gauge Theory of Elementary Particle Physics [9],

F.Yndurain:Quantum Chromodynamics [10],

Y.Dokshitzer,V.Khoze,A.Mueller and S.Troyan:Basics of QCD [11],

G.Altarelli:Partons in Quantum Chromodynamics [12],

R.Field:Quantum Chromodynamics [13],

K.Huang:Quarks and leptons [14],

F.Halzen,A.Martin:Quarks and leptons [15],

C.Quigg:Gauge theories of strong,weak and electromagnetic interactions [16].

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Although this text concerns primarily the strong interactions,some working knowledge of the current electroweak theory,going usually under the name Standard Model(SM),is necessary.I have therefore included elements of this theory on several appropriate places,but the reader is recommended to consult the new book of Jiˇr′?Hoˇr ejˇs′?Fundamentals of electroweak theory[17]for excellent pedagogical exposition of the foundations of modern gauge theory of electroweak interactions.This book also gives the reader some feeling how nontrivial and twisted was the development of its basic concepts and how vital was the permanent confrontation between emerging theory and experiment.

The text has been written with two types of students in mind.First,a beginner in the?eld would probably like to concentrate on mastering the quark-parton model and have only limited aspirations as far as QCD is concerned.An eager experimentalist could be a typical example.On the other hand,for theoretically oriented or postgraduate students more emphasis on the QCD would be appropriate.To reconcile both of these needs the text is written at two levels:elementary and advanced ones.Parts of the text contain the advanced topics are marked in the Contents as well as in the text by an asterisk.

Helped by comments and suggestions of a number of my students,I have tried to correct misprints and errors,but more are likely to remain.I will therefore be grateful to the reader for alerting me2to them or for any other comments or suggestions concerning the content of this text

Beside minor corrections and modi?cations of the previous versions,the present one contains one im-portant extension,concerning the the emphasis on historical aspects of major experimental discoveries and their in?uence on the origins of basic theoretical concepts.The more I know about this point,the more I am convinced that without the knowledge of historical circumstances,with all its turns,of the development of the Standard Model,we cannot appreciate the complicated path to its current status and the nontrivial nature of its basic structure.I have therefore added to most of the Chapters introductory Section containing basic information on the history relevant for the subject dealt with in that Chapter.On a number of places, particularly in the Chapter on deep inelastic scattering and the origin of the Parton Model,some of such remarks are inserted directly into the text as well.In these historical digressions I have greatly bene?ted from several books of Abraham Pais,particularly his remarkable historical treatise Inward bound[18],which contains wealth of information unmatched by any other source.The Czech reader may?nd an excellent description of historical development of the Standard Model with emphasis on the electroweak sector in[19].

Many important topics that might be relevant for students coming to the rapidly expanding?eld of subnuclear physics had to be left out,either because of my incompetence,or the lack of space.In a future version I intend to include introductory expositions concerning

?Basic ideas and techniques of the so called Monte-Carlo event generators.These programs,based on current understanding of the Standard Model and generating,via the Monte Carlo techniques,the whole complicated event structure,have become indispensable tools not only in the theory,but even more in experiment.All modern particle detectors use them heavily at various stages of the extraction of meaningful information from raw experimental data.

?The procedure of extracting parton distributions functions from the so called global analyses of exper-imental data on hard scattering processes.

?Theoretical ideas and experimental information concerning the concept of the structure of the photon.

In writing up this text I have bene?ted from innumerable discussions with my colleague and friend P. Kol′aˇr,which helped shape my understanding of the subjects covered in these lecture notes.The section on dimensional regularization is based to large extent on his notes.I am grateful to Jiˇr′?Hoˇr ejˇs′?for initiating my interest in historical aspects of the development of the Standard model in general and QCD in particular.

I have enjoyed greatly many interesting discussions we have had over the last year concerning the birth of fundamental concepts of this theory as well as of the crucial discoveries leading to current picture of subnuclear structure and forces.We share the point of view that good understanding of these aspects of the Standard Model is vital for its further development.

Prague,February2007

2Email:chyla@fzu.cz

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Chapter 1

Introduction

1.1What are these lectures all about?

Before going into the systematic exposition of the quark-parton model and Quantum Chromodynamics,I will brie?y review the basic concepts and overall structure of the Standard model (SM),the theoretical framework that encompasses the current status of our understanding of strong and electroweak interactions.Clear and pedagogical introduction into the electroweak sector of the Standard Model can be found,for instance,in [17]).

The fundamental objects of the SM are fermions with spin 1/2,pided into two distinct groups:quarks and leptons ,forming three generations of identical structure.Each of the 6quarks carries a distinct internal quantum number,called ?avor ,which is conserved in strong (but not electroweak)interactions.The corresponding antiparticles are grouped in a similar pattern.The interactions between these fundamental fermions are mediated by the intermediate vector bosons (W ±,Z,γ)in the case of electroweak interactions and by gluons in the case of strong interactions.The full set of fundamental particles of the SM is completed by the so far elusive Higgs boson .Gravitational e?ects are neglected.In this text we shall be interested primarily in the dynamics of strong interactions,but leptons turn out to be very useful tools for the study of the structure of hadrons ,particles composed out of quarks and gluons.

The major conceptual problem in QCD is Generations charge 1. 2. 3.Quarks 2/3u(up)c(charm)t(top)-1/3d(down)s(strange)b(bottom)Leptons 0νe νμντ-1e ?μ?τ?Table 1.1:Fundamental fermions of the Standard model related to the fact that although we talk about quarks and gluons as basic building blocs of matter,they have so far not been observed in the nature as free particles.This funda-mental property of QCD,called quark and gluon con?nement makes it fundamentally

di?erent from the familiar Quantum Electro-dynamics.I shall discuss our current under-

standing of this phenomenon in the part on hadronization,but it is clear that it has also far-reaching philosophical implications.

The reverse side of the quark (or better,color,con?nement)is another novel phenomenon,called asymp-totic freedom .It turns out,and will be discussed extensively later on,that as quarks and gluons get closer to each other,the strength of their interaction,measured by the e?ective color charge,progressively decreases until it vanishes at zero distance.Both of these surprising phenomena,i.e.quark con?nement as well as asymptotic freedom,turn out to be related to the nonabelian structure of QCD,or,in simpler terms,to the fact that gluons carry color charge (contrary to photons,which are electrically neutral).1With color charge introduced,the basic set of quarks in Table 1.1should actually be tripled as each of them exists in three di?erent color states.

In order to understand the way the current knowledge of the structure of matter has in the past century been discovered by colliding various particles,it is vital to understand in detail the setup and reasoning

1This

relation is,however,sensitive to several aspects of the theory,like the numbers of colors and ?avors and the dimen-sionality of the space-time.For instance,in 2+1dimensions the con?nement is the property of classical theory already,while even in nonabelian gauge theories asymptotic freedom may be lost in the presence of large number of quark ?avors.7

used by Rutherford in1911in his discovery of atomic nucleus.In fact,all the major discoveries concerning the structure of matter,which are the subject of these lectures,repeat the same basic strategy pioneered by Rutherford and di?er only in conclusions drawn from experimental?ndings.I will therefore start by recalling the essence of his experiment and then discuss some important aspects of the scattering process,?rst in classical and then in nonrelativistic quantum mechanics.

1.2Rutherford experiment

The schematic layout of the Rutherford experiment is sketched in Fig. 1.1.Narrow beam ofα-particles emanating from the natural radioactive source inside the lead box and formed by means of a diaphragm had been directed towards a thin golden scattering foil.Scatteredα-particles were observed as scintillations on the screen made of ZnS.Counting the number of such oscillations per unit of time in a given interval of scattering angles allowed Rutherford and his collaborators Geiger and Marsden to measure the angular distribution of the scattered particles.According to the”pudding”model of the atom,popular at the beginning of the last

Figure1.1:Layout of the Rutheford experiment and prediction of the“pudding”model of the atom. century,the positive electric charge was assumed to be distributed more or less uniformly over the whole atom.The electrons,known since1897to be part of the atom,were imagined to resemble raisins in the 0a9ebc68af1ffc4ffe47acafing the classical mechanics to calculate the angular distribution ofα-particles scattered in such a medium(for details of this calculation see next Section)lead to the expectation,sketched in Fig.1.1c,that almost all the particles scatter at very small angles,whereas the number of particles scattering at large angles should be entirely negligible.Such distribution is in marked contrast to what we expect for the scattering o?a pointlike charge of the same magnitude,described by the Coulomb potential V(r)=α/r.The latter leads to angular distribution proportional to1/sin4(?/2),which peaks at small angles,but is nonnegligible even at very large angles including those corresponding to backscattering.

Surprisingly,the observations made by Geiger and Marsden and represented in a schematic way in Fig.

1.2a,showed convincingly that the large angle scattering is quite frequent,in complete disagreement with the expectations of the“pudding”model of atom.Moreover,quantitative analysis of the angular distribution of scatteredα-particles suggested that it is proportional to1/sin4(?/2),the form expected for the scattering o?a pointlike electric charge.The conclusion drawn by Rutherford(see next Section for details of the reasoning)

Figure1.2:Results of the Rutheford experiment and their interpretation as evidence that positive of the atom charge must be concentrated in a small spatial region.

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Figure 1.3:Schematic layout of the apparatus used by Geiger and Marsden (left)and the Table containing the results of their measurements.

was that the positive electric charge is concentrated in atom inside the region that is much smaller than the atom itself,see Fig.1.2c.Since the experiments at SLAC in 1950-1960,to be mentioned in Chapter 4,we know that the nucleus is actually not pointlike,but has a ?nite radius,which is about 100000times smaller than that of the hydrogen atom.The concept of atomic nucleus was born.

In view of the pioneering character of the Rutherford experiment it is perhaps useful to recall two important circumstances.First,the “Rutherford experiment”was actually a series of observations carried out by Geiger and Marsden between 1906and 1913,which had lead Rutherford to his model of atom.In Fig.

1.3,taken from the original paper of Geiger and Marsden [20],the vertical cross section of their apparatus is shown.Its main components were a cylindrical metal box B the size of a present day desktop PC,which contained the source R of α-particles,the scattering foil F,and a microscope M,to which the ZnS screen S was rigidly attached.The microscope M and box B,fastened to a platform A,could be rotated around the foil and radioactive source,which remained in position.The beam of α-particles from the source R formed by the diaphragm D was directed onto the scattering foil F.By rotating the microscope the α-particles scattered at di?erent angles could be observed on the screen S.The distances between the source R and the foil F and this foil and the screen S were both typically 2cm.Observations were taken at scattering angles between 5?and 150?.Geiger and Marsden measured the angular dependence of a number of scintillations observed during their experiments and compared it to the form 1/sin 4(?/2)expected for the pointlike positive electric charge.The arrangement in which the microscope rotates around the stationary beam and target,invented by Geiger and Marsden,had since been used in modern variants in many other experiments,most notably at SLAC in ?fties and sixties (see Chapter 5for details),and usually called “single arm spectrometer”.s In the paper [20],which includes the comprehensive analysis of all technical aspects of their experiments Geiger and Marsden presented their results in a form of the Table,reproduced in Fig.1.3,which lead them to conclude “considering the enormous variation in the numbers of scattered particles,from 1to 250000,the deviations from constancy of the ratio are probably well within the experimental error.The experiments,therefore,prove that the number of α-particles scattered in a ?nite direction varies as 1/sin 4(?/2).”The Summary of [20]starts with the words:“The experiments described in the foregoing paper ···there exists at the center of the atom an intense highly concentrated electrical charge”as proposed by Rutherford.All scattering experiments perfomed ever since are in essence nothing but more or less sophisticated versions of the Rutherford experiment.What has,however,changed quite dramatically,are the characteristics of fours basic ingredients of any scattering experiment:

?source of scattering particles,

?nature and form of the target,

?technique for detecting the scattered particles,

?framework for theoretical predictions to be compared to data.

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Instead of simple natural radioactive source physicists now use powerful accelerators,where protons,antipro-tons,electrons and positron are given energies bilion times bigger that those in Ruthherford experiment. The role of a simple golden foil has been taken over by sophisticated arrangements of the target material and the task of observing and recording the position of the scattered particle,entrusted to the eyes and hands of Geiger and Marsden,requires now huge detectors the size of a big family house and weighing thousands of 0a9ebc68af1ffc4ffe47acafst,but not least,also the complexity of theoretical calculations has increased enormously.Instead of a simple analytic expression,like the formula(1.4in the next Section),theoretical predictions that can be compared to experimental data employ in most cases complicated computer programs.

1.3Bohr model of atom

Rutherford’s discovery of the atomic nucleus lead directly to the formulation of the so called planetary model of atoms,in which negatively charged electrons orbited positive,practically pointlike nuclei,in much the same way as planets orbit the Sun.Since the very beginning this model faced serious problems to explain the basic features of atoms,in particular

?the existence of only discrete sets of allowed classical orbits characteristic for each element,

?the stability of the lowest energy(ground)states and

?the identity of all atoms of a given element.

By Maxwell laws of classical electrodynamics,negatively charged electrons orbiting positive nuclei should copiously radiate electromagnetic waves and thereby rapidly collapse,which,fortunately,does not happen. Equally incomprihensible apeared the fact that only well de?ned,discrete classical orbits were allowed to reproduce the obseved spectrum of deexcitation lines.Nothing like that holds for planetary systems hold together by gravity.A collision with an asteroid would cause arbitrarily small perturbation of an Earth’s orbit,kicking the Earth into a new orbit,where it would live until the next encounter.

The impossibility to reconcile the laws of classical electrodynamics with the basic properties of atoms lead Bohr to the formulation of his model of atomic structure.This model attempted to explain the above meantioned properties of atoms by adopting several quantum postulates.and employing the hypothesis of energy quantization,laid out by Planck in1900in order to explain the spectrum of blackbody radiation. The resulting“Old Quantum Theory”was essentially a desperate attempt to apply the old classical concepts of particle tracectory for the description of subatomic world.It was developped into a rather sophisticated framework which successfully reproduced the above basic properties of atoms,by Bohr,Sommerfeld and others.It eventually failed but to understand the reasons for this failure is essential for understanding the fundamental inevitability of abandoning the classical physics for the description of the microworld.There is no space in this text to elaborate on the Bohr Model,but it is still worthwhile to read the original Bohr papers[22,23]where its foundations are laid out.

1.4Scattering experiment and the concept of cross-section

As most of the measurements in the world of elementary particles are based on the scattering experiments let me?rst recall some basic relevant facts.They are simple but may get lost in the?ood of mathematical formalism on one side and practical realization of such experiments on the other.

Consider?rst the scattering of a pointlike particle by a(for simplicity spherically symmetric)potential V(r),?rst in nonrelativistic classical mechanics.The familiar example of how physical laws can be inferred from experimental observations is the story of Johannes Kepler,who by observing the movement of celestial objects was able to formulate his famous laws.This procedure can be formalized as follows.The beam of particles is sent on the force center we want to study and by observing the resulting de?ection we can, if the beam is sent under all impact parameters b,(see Fig. 1.4)deduce most of the information on the potential.I used the term“most”as there are certain features,concerning the properties of bound states and resonances in quantum physics,which can’t be determined in this way.Moreover,as we shall see in the next Section,to probe the form of the potential at all distances requires,in classical physics strictly and in quantum physics practically,in?nite energies.Instead of the whole classical path it,in fact,su?ces to measure the distribution of the scattering angle?(b)as a function of the impact parameter b.

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Experimentally,the above algorithm can be realized by sending a uniform ?ux of test particles,de?ned by their number J per unit area perpendicular to the incoming beam,onto the target,and measure the angular distribution N (?)d cos ?,the number of particles scattered into the angular interval (?,?+d cos ?).In this approach the impact parameter b does not appear in the ?nal expression,but we have to ensure that the ?ux of incoming particles is uniform over the whole plane perpendicular to its direction.This leads us to the de?nition of the cross-section ,here di?erential with respect to the scattering angle ?

d σ(?,φ)≡N (?,φ)d?J ?d σd?=N (?)J =N (?)2πJ .(1.1)

Provided there is one-to-one correspondence

between

Figure 1.4:Layout of the scattering experiment.the impact parameter b and scattering angle ?the

di?erential cross section (1.1)can expressed in terms

of b (?)as follows

d σd?=b (?)d b (?)d cos ?

(1.2)where d?=d cos ?d φwith ?,φbeing the polar and

azimuthal angles respectively.Note,that although

the scattering angle ?is a function of the impact

parameter b the opposite is not necessarily the case as more impact parameters may lead to the same scattering angle.This fact will be crucial in the analysis,performed in the next Section,of mathematical formalism behind the interpretation of the Rutherford experiment.

For the scattering of a unit test charge on the Coulomb potential V (r )=α/r corresponding to unit (in units of positron charge)in?nitely heavy target charge,the relation between the impact parameter b and the scattering angle ?reads

b (?)=1κtan(?/2),κ≡p 2mα=2E α???(b )=2arctan 1κb =π?2arccos 1√1+κ2b 2

,(1.3)which leads to the Rutherford formula for di?erential cross section

d σd?=14κ2sin 4(?/2)=α2m 24p 4sin 4(?/2)=α216E 2sin 4(?/2)=4α2m 2q 4,(1.4)

where m,p are the mass and momentum of the beam particle and q 2=4p 2sin 2?/2is square of the momentum

transfer q ≡ p ? p

.The derivation of (1.3-1.4),or in general of the di?erential cross section for scattering in a spherically symmetric potential is quite simple.As the particle scattering on a spherically symmetric potential V (r )moves in a plane it su?ces to work in two dimensions.Writing the corresponding lagrangian in spherical coordinates we get L =12

m (˙r )2+r 2(˙φ)2 ?V (r )(1.5)where the dot stands for time derivative.This lagrangian implies two constants of motion:orbital momentum M and energy E

M ≡mr 2˙φ,E ≡12m (˙r )2+U (r ),U (r )≡M 22mr 2

+V (r ),(1.6)which can be expressed in terms of initial momentum p and impact parameter b as M =pb,E =p 2/2m .The scattering angle ?equals ?=π?2φ0where φ0is given as

φ0≡

∞

r 0d φd r d r = ∞r 0˙φ˙r d r = ∞r 0M mr 2d r 2(E ?U (r ))/m (1.7)

where we have simply substituted for the time derivatives the expressions following from (1.6),and r 0,denoting the smallest distance of the trajectory from the scattering center,characterized by the vanishing

11

of time derivative ˙r =0,which implies that it is given as the solution of equation E =M 2/2mr 2+V (r ).Solving the equation for r 0and evaluating (1.7)for the Coulomb potential V (r )=α/r yields (1.3)and

r 0=1κ 1+ 1+κ2b 2 =1κ 1+ 1+cot 2(?/2) ≥r min ≡2

κ=αE .

(1.8)To penetrate close to the origin requires large angle scattering,but the angular dependence r 0(?),plotted

in Fig.1.5,shows that above roughly ? 120?the values of r 0is insensitive to ?and equals approximately 2/κ.The angular distribution of scattered α-particles as measured by Marsden and Geiger [20],reproduced in Fig.1.5,showed excellent agreement with the 1/sin 4(?/2)behavior expected of a pointlike electric charge throughout the covered region,which extended up to 150degrees.Taking into account the energy of α-particles from radioactive decay of radium was 7.68MeV,its electric charge (2),as well as that of gold (79)

gave

r min .

=30fm,which is about 10000times smaller than the radius of atom.

One feature of the formula (1.4)is worth not-Figure 1.5:κr 0(?)(left)and the angular distribution of α-particles as measured by Geiger and Marsden (right).

ing.Although (1.4)decreases for a ?xed scat-tering angle ?with increasing momentum p of the incoming particle,the form of this distribu-tion remains the same for all momenta p .This follows immediately from the relation between ?and b :to get the same scattering angle ?as p increases,one must simply shoot the incom-ing particle at appropriately smaller impact parameter!This form invariance of the dif-ferential cross section does not,however,hold generally.It is obvious that for a ?nite incom-ing momentum p the angular distribution is

sensitive to the form of the potential V (r )at distances r ≥r 0only.The value of r 0depends

on p and m as well as details of the potential V (r )in the vicinity of r =0.To probe V (r )at in?nitely small distances requires in?nitely large energies of incoming particles.This statement holds in classical,but as we shall see below,not in quantum mechanics.

The integrated (over the angles)cross-section perges due the behavior of (1.4)at small angles.In classical physics and as far as integrated cross-sections are concerned,there is no principal di?erence,between the Coulomb potential α/r and,for instance,the familiar Yukawa potential (α/r )exp(?μr )as both are in?nite.In quantum physics the situation is di?erent as there the Yukawa,but not the Coulomb,potential has ?nite integrated cross-section.

In classical physics the concept of cross section is useful,but may be dispensed with as more direct methods,based on the classical equations of motion for particle trajectories,are available.In quantum physics,on the other hand,this concept plays an indispensable role,as the latter methods inapplicable.We can not trace down exact trajectories of quantum particles (indeed they have no good meaning at all)neither can we determine impact parameter of each incoming particle inpidually.

1.5Scattering in nonrelativistic classical physics

Let me now discuss in some detail the scattering of a pointlike test particle of unit charge on a repulsive potential V (r )generated by the spherically symmetric distribution ρ(r )of unit electric charge inside the sphere of radius R .As no charge is located outside this sphere,the Gauss theorem tells us that the force

(and thus the intensity of the electric ?eld E

)outside the sphere coincide with that generated by unit pointlike charge.For the corresponding potential this implies the following relation

r ≥R :V (r )=αr

,(1.9)

r ≤R :

V (r )

=

e (r )α

r

+4πα

∞

r

zρ(z )d z,e (r )≡4π

r

z 2ρ(z )d z,

d V (r )

d r =?

e (r )αr

2(1.10)where e (r ),the e?ective charge at distance r ,is just the total electric charge located inside the sphere of

radius r .Clearly,e (R )=1.

12

Imagine we want to ?nd out the form of ρ(r )from the results of a scattering experiment by measuring the angular distribution of scattered particles.It is obvious that to do that the test particle must penetrate closer to the source of the potential than R ,otherwise it will not be in?uenced by the form of ρ(r )at all and will feel the total charge e 0only.This requires that κ≡p 2/mα>κmin =2/R .Even then,however,not all particles will pass through the region r

I will now consider beside the pointlike charge three di?erent forms of extended charge distribution ρ(r )that will illustrate the relation between the structure of the target and the corresponding angular distribution of scattered particles.First two of them coincide at r ≥R with the potential α/r induced by the pointlike charge distribution and di?er only inside the sphere r ≤R .The third distribution has no sharp edge but the parameter R characterizes the rate of exponentially decreasing factor exp(?r/R ).We ?nd (x ≡r/R )ρ1(R,r )=δ(1?x )4πR 3,e 1(x )=θ(x ?1),V 1(r )=αr [θ(x ?1)+xθ(1?x )](1.11)ρ2(R,r )=34πR 3θ(1?x ),e 2(x )=x 3θ(1?x )+θ(x ?1),V 2(r )=αR 32?x 22 θ(1?x )+αr θ(x ?1)(1.12)ρ3(R,r )=18πR 3e ?x ,e 3(x )=1?e ?x (1+x +x 2/2),V 3(r )=αR e 3(x )x

+e ?x (1+x ) (1.13)Note that the ?rst distribution ρ1(R,r )corresponds to putting the total (in our case unit)charge uniformly on the surface of the sphere with the radius R ,whereas ρ2(R,r )corresponds to constant volume charge density inside that sphere.The former will play important role in quantum ?eld theory when introducing the concept of renormalized coupling constants.

The form of the potentials V i (r ),shown in Fig. 1.7,inside the sphere leads to the modi?cation of the relation (1.3)that holds for the pointlike source.For the potential V 1(r )we ?nd after elementary calculation ?(R,κ,b )=π?2arccos 1√1+κ2b 2 ,b >b 0=1κ (Rκ?1)2?1,(1.14)?(R,κ,b )=π?2 arccos 1√1+κ2b 2 +arccos b b 0

?arccos 1+κb 2/R √1+κ2b 2 ,b ≤b 0.(1.15)Graphical representation of this relation is shown in Fig.1.6for R =

1

Figure 1.6:a)?(R,κ,b )as a function of b for the distribu-tion ρ1(1,r )and several values of κ(solid curves),from above in decreasing order.The dotted curves correspond to R =0.(in arbitrary units)and ?ve values of the parameter κ,one of them,

κ=2=κmin ,corresponding to the threshold case that the scattered

particles do not enter the interior of the sphere with radius R and are

thus not in?uenced by the potential inside it.Several conclusions can be

drawn from this ?gure.

First,for reach κ>κmin there is a maximal scattering angle ?max (R,κ)=

?(R,κ,b 0)that occurs for b =b 0and is a decreasing function of κ.This

behaviour simply re?ects the fact that large scattering angles require

strong repulsive potential,which,however,is not the property of the po-

tential V i (r ).The fact that the angular distribution vanishes for ?>?max

is a direct consequence of the sharp edge of the distribution ρ1(r ).For

smoothly vanishing ρ(r )the resulting angular will not vanish at large

angle but will merely be strongly suppressed there.

Secondly,to calculate the angular distribution from the known depen-dences ?(R,κ,b )we cannot simply invert then and use (1.1)as there are two impact parameters for each scattering angle ?.We must therefore use (1.1)separately in the two regions b b 0and then sum the results.As a result,the angular distribution will di?er from that of the pointlike charge distribution not only for ?>?max (R,κ),where it vanishes,but also in the region ?

Third,as for ?xed κthe maximal angle ?max (R,κ)is a decreasing function of the radius R ,we have to go to large angles,i.e large momentum transfers,to probe the charge distribution inside the sphere of that radius.

13

Figure1.7:The form of potentials V k(r)de?ned in(1.11-1.13)and multiplied by R/α(a),corresponding to the e?ective charges e k(r/R)plotted in(b)and the square of the formfactors F k.Dotted curves correspond to the pointlike charge at the origin.

We have thus seen that by measuring the angular distribution su?ciently precisely we can determine the form of the potential and thus the corresponding charge distribution inside the sphere.In principle we don’t have to go to large angles to do as the di?erences at large and small angles are related.We have,however, use su?ciently energetic primary particles in order to satisfy the conditionκ=p2/mα>κmin=2/R.In quantum theory,as we shall see below,the?rst statement holds as well,but the second does not.

So far we have treated the distributionsρ(r)of the target charge as?xed,whereas in realistic conditions of the Rutherford and other experiments the corresponding carriers were actually other particles.This is expected to be a good approximation,if the rest mass of these particles is much larger than the momentum transfer q,which,indeed was the case in the Rutherford experiment.

1.6Scattering in nonrelativistic quantum physics

In nonrelativistic quantum mechanics the di?erential cross section dσ/d cos?for elastic scattering of a spinless pointlike particle with mass m and a unit charge2on a spherically symmetric potential V(r)is given(in suitable normalziation)as square of the corresponding scattering amplitude f(?).In the lowest order perturbation theory,called usually the Born approximation,f(?)is just the Fourier transform of the potential acting on the scattering particle:

dσd?=|f(?)|2,f(?)=?

m

2π

exp(i q r)V(r)d r, q≡ p? p ,q2=4p2sin2?/2.(1.16)

For the Coulomb potential of the form V(r)=α/r we easily?nd

f(?)=f(q2)=2αm

q2

?dσ

Born

d?

=

4α2m2

q4

=

α2m2

4p4sin4(?/2)

(1.17)

The form(1.17)is often taken as de?ning the pointlike nature of the target particle,generating the potential V(r).Remarkably,the Born approximation in quantum mechanics coincides for the Coulomb potential with exact result of the classical calculation.This coincidence does not,however,have a more general validity. For Yukawa potentialαexp(?r/R)/r we similarly?nd

f(?)=f(q2)=

2αm

(q2+μ2)

?dσ

Born

d?

=

4α2m2

(q2+μ2)2

=

4α2m2

(4p2sin2(?/2)+μ2)2

(1.18)

where the presence ofμ≡1/R in the numerator of(1.18)screens o?the singularity of the Coulomb amplitude (1.17)in forward direction so that the integrated cross-section becomes?nite.

σBorn integ =

16πα2m2

4p2+μ2

1

μ2

(1.19)

2For beam particle with charge e the results can be obtained by the substitutionα→αe.

14

For the case the potential V (r )is generated by the (spherically symmetric)distribution ρ(r )of elementary sources,each of them having the same form V (r ),i.e

V ρ(r )≡

d z ρ(z )V ( z ? r )(1.20)

Born approximation for the scattering amplitude f ρ(q 2)yields,upon exchanging the order of integrations,

f ρ(q 2)=F (q 2)f (q 2),F (q 2)=

exp(i q r )ρ( r )d r ,(1.21)

where the elementary scattering amplitude f (q 2)is given in (1.16).For the distributions ρk (r )corresponding to the potentials V k (r )de?ned in the previous Section,the form of their formfactors F k (q 2)is shown in Fig.

1.7.The rapidly dying oscillations at large angles (0a9ebc68af1ffc4ffe47acafrge values of qR )for the case of charge distributions ρ1and ρ2re?ect their sharp edges.No such oscillations are,on the other hand,present for the smoothly decreasing distribution ρ3.Note that all the formfactors F k (qR )deviate from unity,which corresponds to the pointlike charge distribution,at all qR .This implies that contrary to classical mechanics,there is no strict lower limit on energy E of the incoming particle necessary for probing the charge distribution inside the sphere R .On the other hand,to access the region where the formfactors F k di?er appreciably from unity,requires qR 1.This in turn implies pR 1and thus sets the lower limit on p .

1.7Scattering on bound systems

So far we have discussed scattering o?a given distribution of elementary scatterers described by some continuous function ρ(r )normalized to the unity.To come closer to real situation let us now consider a

bound system of such elementary scatterers,described by potentials V i (

r ? r i )=αi V ( r ? r i ),still ?xed at points r i and di?ering only by the magnitude of their “charges”αi ,which sum up to the total charge of the target i αi ≡α,which can be without loss of generality set to unity.Let us moreover assume the existence of a minimal distance ?min between any pair of them.In the Born approximation the expression for the scattering amplitude F (?)on such a composite system reads

F (?)=?m 2π exp(i q r ) i αi V ( r ? r i )d r = i αi e i q r i ?m 2π exp(i q r )V (r )d r f (?)

(1.22)where the last integral is just the elementary amplitude for the scattering on a pointlike centre with unit charge.The di?erential cross-section for the scattering on such a composite target is then given as

d σcomp d?=d σpoint d? i αi exp(i q r i ) 2=d σpoint d? i α2i +2R

e i

(1.23)This expression has simple forms in two di?erent regions:?q 2→0:where Z → i

that on the pointlike target with the unit total charge α.For small momentum transfers,the beam particle “sees”merely the total charge of the target.

?q 2→∞:where Z →0(due to nonzero ?min !)and we get

d σcomp d?→d σpoint d? i α2i = i d σi d?(1.24)

In this case the beam particle “sees”the inpidual charges and measures the incoherent sum of inpidual cross-sections,proportional to squares of the charges.Assuming there are N scattering centers each carrying the same charge 1/N ,we conclude that the di?erential cross section (1.24)decreases at large q 2with N like 1/N !

15

Realistic systems,like atoms or nuclei,their constituents are not?xed in space,but bound by some,usually

two-body,forces and therefore we have to take into account these forces when calculating scattering on

inpidual scattering constituents.Provided the system is weakly bound,i.e.the characteristic binding energy is?nite and small with respect to momentum transfer q of the collision,this binding can to?rst

approximation be neglected and we can again use(1.24),although the kinematics must take into account

the recoil of the target constituent.The main di?erence with respect to?xed scattering centers will be in

the identity of the?nal state.It is obvious that a violent scattering on a bound system will usually break

it up and thus lead to inelastic collision.The probability that after such violent collision the constituents

will recombine into the original system,though nonzero,is very small.So to see inside the bound system

we thus have to break it up into the inpidual constituents!

The above reasoning is straightforward and almost trivial for weakly bound systems,but can under certain circumstances be used even for the strongly bound ones.Imagine,for instance the system of particles bound

pairwise by the linearly rising potentials V(r)=ωr,which prevents the separation of inpidual particles with any?nite energy.On the other hand,provided that q2 ω,which guarantees that the interparticle distances remain small during the interaction time,estimated by1/q,compared to the distances where the

linearly rising potential becomes comparable to q,we can separate the process into two stages:

Hard scattering:describing the scattering of incoming particles on inpidual scatterers treated as free particles,which is sensitive to their charges,

Recombination:nontrivial process in which the excited system de-excites by producing the original or other strongly bound systems.

The essential feature of this space-time separation of the hard inelastic collision is that the recombination

stage determines the?nal states of the collision,but not the angular distribution of the scattered test particles!

1.8Exercises

1.Solve Gauss’law?φ( r)=κδ(r)in1+1and2+1dimensions and discuss the results.

2.Derive(1.2).

3.Derive(1.4).

4.Derive(1.18)and(1.19).

5.Evaluate the formfactor(1.21)as well as the potential(1.10)for the distributionρY(r)=(μ3/8π)exp(?μr)/κ.

16

Chapter2

Elements of groups and algebras

2.1De?nitions and basic facts

Only a brief reminder of the very basics of group theory is given in this chapter.The emphasis is put on practical abilities to carry out simple calculations that will be needed in the following chapters when discussing the quark-parton model and QCD.

Group:a set G with binary operation“?”satisfying the following conditions:

1.?x,y∈G:x?y∈G

2.?e∈G:?x∈G,e?x=x?e=x(existence of a unit element e)

3.?x∈G?x?1∈G:x?x?1=x?1?x=e(existence of an inverse element)

4.?x,y,z∈G:x?(y?z)=(x?y)?z(associativity of?)

Groups enter physics basically because they correspond to various symmetries.The concept crucial for the description of transformations of physical quantities under these symmetry transformations is that of the group representation.

Group representation:the mapping D:G?L H of the group G onto the space L H of linear operators on Hilbert space H,which preserves the property of group multiplication“?”

?x,y,z∈G:x?y=z?D(x)D(y)=D(z)(2.1) where the multiplication on the r.h.s.of(2.1)is de?ned in the space L H of linear operators on H.The representations will in general be denoted by boldface capital D with possible subscripts or superscripts,or as is common for SU(3)group,by a pair of nonnegative integers(i,j),specifying the so called highest weight of the representation(see Section2.6).The image of a group element g∈G in such a representation will be denoted simply as D(g).

Examples:

?G=R,the set of real numbers with conventional addition as the group binary operation?,D(x)= exp(iαx)whereαis a?xed real number.This representation plays a crucial role in the construction of abelian gauge theories,like QED.

?P3,permutation group of three objects.De?ne(1,2)as permutation of1,2etc.,(1,2,3)as simultaneous transpositions:1→2,2→3,3→1and e as a unit element.One of its representations is given by the following six matrices,with normal matrix multiplication de?ning the product of linear operators:

D(12)=?

?

010

100

001

?

?,D(13)=

?

?

001

010

100

?

?,D(23)=

?

?

100

001

010

?

?,

17

D (123)=??0

011

00010??,D (321)=??010001100??,D (e )=??100010001

??,Realization of operators by matrices:the action of any linear operator ?O

on vectors |j from a given normalized basis of H can be represented by means of the matrix O ij de?ned as

O ji ≡ j |?O |i ??O |i =O ji |j ,(2.2)

where the summation over the repeating indices is understood as usual.In all the following applications I shall,when talking about the group representation,always have in mind the above matrix representation.Equivalence of representations:takes into account the fact that the matrices O ij depend on the chosen basis of H .Two representations D 1and D 2are said to be equivalent if:

?S ∈L H :?x ∈G ,D 1(x )=SD 2(x )S ?1.(2.3)

The transformation of D (x )can be interpreted as resulting from the change of the basis of H induced by S .Direct sum of representations:assuming D i act on corresponding Hilbert space H i of dimension n i ,i =1,2,the direct sum H =H 1⊕H 2is the Hilbert space of dimension n 1+n 2and its n 1+n 2dimensional basis spanned by the set of vectors

|e 11 ,|e 12 ,···,|e 1n 1 basis of H 1,|e 21 ,|e 22 ,···,|e 2n 2 basis of H 2.(2.4)

The corresponding direct sum of representations D =D 1⊕D 2is made up from matrices (n 1+n 2)×(n 1+n 2)of the block diagonal form D 100D 2

.(2.5)Direct product of representations:a more complicated but simultaneously more interesting 0a9ebc68af1ffc4ffe47acafing the same notation as above,the basis of the direct product H =H 1?H 2is formed by n 1×n 2pairs of vectors of the form |e 1i |e 2j where |e j i ∈H j ,j =1,2.The direct product of representations D 1,D 2,D =D 1?D 2is formed by the matrices which act on these basis vectors as follows (g ∈G is an element of group G )D 1?D 2(g )

∈D 1?D 2|e 1i |e 2j =(D 1(g ) ∈D 1|e 1i )(D 2(g ) ∈D 2

|e 2j ).(2.6)The matrices corresponding to (2.6)are (n 1×n 2)×(n 1×n 2)dimensional and can be written as

[(D 1?D 2)(g )]ij,i j =[D 1(g )]ii [D 2(g )]jj .(2.7)

Reducibility of a representation:the possibility to transform all elements of a given representation D ,acting on the Hilbert space H ,by means of a unitary operator S to block–diagonal form:?S ∈L H :?x ∈G ,SD (x )S ?1= D 1(x )00D 2(x ) ,(2.8)where the matrices D 1(x ),D 2(x )act on Hilbert spaces H 1and H 2respectively and H =H 1⊕H 2.If such a unitary operator S does exist the representation D is called reducible and can be written as a direct sum D =D 1⊕D 2.If not D is said to be irreducible .

Further important concepts:

?Abelian group:the group multiplication ?is commutative .

?Finite and in?nite groups:according to the number of independent elements

?Compact and noncompact groups:roughly speaking compact groups have ?nite “volume”as measured by some measure on the group space,while the noncompact have in?nite volume.Example:the group of phase factors introduced above is compact,as the unit circle in complex plane is the ?nite.

18

2.2Lie groups and algebras

I shall not attempt to give an exact de?nition of this important concept for the case of a general Lie group, as it would require introducing too much of heavier mathematical arsenal,which we shall actually not need in our practical applications.Instead let me emphasize the basic attributes of this class of groups.Lie groups are characterized by the fact that their elements are labelled by a set of continuous parameters,which can be imagined as coordinates in a subset of R n,n-dimensional Cartesian space with conventionally de?ned metric and topological properties.These properties are necessary for the formulation of the concept of continuous mapping and continuous functions,which are central to mathematically rigorous de?nition of Lie groups. The exact meaning of the words“labelled by a set of continuous parameters”is also more involved,but roughly speaking there exists a local one-to-one mapping between the group elements and points in some subset of R n which is continuous in both ways1and which allows us to translate all the operations and questions from the group space to analogous operations and questions in R n where we know what to do. For instance for a continuous group to be a Lie group the group multiplication x?y must be a continuous function of both x and y and the operation of taking the inverse x?1must be continuous function of x. We shall not go into more details of this abstract construction as for us the group elements will always be matrices,parameterized by a few real numbers and for the vector space of matrices the topology is easily de?ned via the norm of a matrix.

For Lie groups,and in particular their matrix representations there is also intuitive understanding of the group“volume”and thus of the di?erence between the compact and noncompact groups.I shall now present,without proofs,several propositions that will be useful in further considerations.

Proposition2.1Any element of a compact Lie group can be written in the form

?g∈G,g=g(αa)=exp(iαa X a),

where the operators X a,called generators of the Lie group G,form the basis of a vector space X(over the ?eld of complex mumbers)of dimension m with the operation of“adding”,denoted as“+”.

The classi?cation and properties of representations are simplest for the class of compact Lie groups,where all those needed in formulating the Standard Model do belong.

Proposition2.2All irreducible representations of compact Lie groups are?nite dimensional. Proposition2.3Finite dimensional representations of a compact Lie group G are equivalent to represen-tations by unitary operators,i.e.the generators X a are hermitian operators.

In the case of matrix groups,or representations,both the elements g of the group G and the generators X a are again matrices.As a result,also the generators of any irreducible representation of a compact Lie group can be represented by?nite dimensional hermitian matrices.Note that the generators of a given Lie group are not determined by this group uniquely,as any change of the basis vectors of the associated Hilbert space implies the change of these generators.

Algebra:A quite general concept,we shall restrict our discussion to algebras over the?eld C of complex numbers.The C-algebra is a vector space A over the?eld of complex numbers equipped with a bilinear binary operation(here denoted as simple multiplication)A×A→A,which means that?x,y,z∈A,a,b∈C ?(x+y)z=xz+yz

?x(y+z)=xy+xz

?(ax)(by)=(ab)(xy)

In the special case of Lie algebra,the binary operation,called“commutator”,is antisymmetric and satis?es the Jacobi identity(2.11).

Structure constants:Due to the fact that the product of elements of a Lie group G like

exp(iλX a)exp(iλX b)exp(?iλX a)exp(?iλX b)

1The continuity,being de?ned by means of open sets,needs topology on the group space as well as in R n.For mathematically rigorous treatment of this topic consult[24].

19

is also an element of this group,it must be expressible as exp(iβc X c).Using the Taylor expansion on both sides we?nd

[X a,X b]≡X a X b?X b X a=if abc X c;βc=?λ2f abc,(2.9) where f abc are real numbers,called the structure constants of the Lie group G.They are by de?nition antisymmetric in?rst two indices.Similarly to generators they are,however,not unique and do depend on the choice of the basis in X.They can be used to express the product of two elements of the group as follows:

exp(iαa X a)exp(iβb X b)=exp(iγc X c)?γc=αc+βc?1

2

f abcαaβb+···(2.10)

Proposition2.4The structure constants f abc satisfy the following Jacobi identity:

f ade f bcd+f cde f abd+f bde f cad=0.(2.11) Proof:(2.11)is a simple consequence of the followin

g relation between the commutators of generators:

[X a,[X b,X c]]+[X c,[X a,X b]]+[X b,[X c,X a]]=0,(2.12) which is straightforward to verify.In this case the vector space is that of the generators X a and the “commutator”is de?ned by the conventional commutator of matrices,representing the generators.Note that the elements of an algebra spanned on the generators of a compact Lie group contain also operators that are not hermitian!

Analogously as in the case of groups we can de?ne representation of an algebra as a mapping of its elements to the operators on some Hilbert space,which conserves the commutator(on a Hilbert space of operators commutator of its elements A,B is simply AB?BA)and introduce the concepts of reducibility, irreducibility etc..

Generators of the direct product D1?D2of n1dimensional representation D1and n2dimensional representation D2are(n1×n2)×(n1×n2)dimensional matrices of the form

X D1?D2

ij,i j

=

X D1

ii

acts on H1only

δjj +δii

X D2

jj

acts on H2only

.(2.13)

Adjoint representation:On the basis of the Jacobi identity it is straightforward to verify that the following n×n matrices,where n is the dimension of the vector space X,

(T a)

bc

≡?if abc(2.14) do satisfy the same commutation relations as X a themselves:

[T a,T b]=if abc T c(2.15) and therefore also form a representation of G.This representation is called adjoint representation and will play an important role in the description of particle multiplets in quark model and of gluons in QCD.

Although we can choose any basis in X,one which is particularly suitable is de?ned by imposing the following normalization conditions:

Tr(X a X b)=λδab,(2.16) where the constantλis positive for compact Lie groups.2In this normalization we have

f abc=?i

λ

Tr([X a,X b]X c)=?

i

λ

Tr([X c,X a]X b)=?

i

λ

Tr([X b,X c]X a),(2.17)

which implies that in this normalization f abc is fully antisymmetric tensor.

Hilbert space of adjoint representation:The Hilbert space associated to the adjoint represen-tation can be constructed from the vector space X spanned on the generators of G by de?ning the binary 2Note that since for compact Lie groups Tr(X a X b)is real symmetric tensor,it can always be diagonalized.

20

operation of a“scalar”product of any two vectors|X a ,|X b ,associated to the generators X a,X b,in the following way:

X b|X a ≡1

λTr

X+

b

X a

,(2.18)

where“+”denotes the hermitian conjugation of the operator and a|,|a are the usual Dirac“bra”and “ket”vectors[24].For the adjoint representation we have the following chain of equalities

T a|T b =(T a)cb|T c =?if acb|T c =|if abc T c =|[T a,T b] ,(2.19) where the?rst equality is a consequence of the de?nition of action of the matrix T a on the ket vector|T b , the second uses just the de?nition(2.14),the third is trivial and the last one returns to(2.15).In other words the action of the operator(for us matrix)T a on the ket vector|T b produces the vector,associated to the commutator of the matrices T a,T b!

Further important concepts:

?Abelian algebra:Algebra with the property that all its elements commute with each other.

?Invariant subalgebra S of algebra A:?a∈S,?x∈A:[a,x]∈S.

?Simple algebra:algebra which contains no nontrivial invariant subalgebra.

?Semisimple algebra:algebra which contains no abelian invariant subalgebra.This type of groups has great physical relevance in theories unifying various kinds of interactions,like the electroweak theory within the SM.

?Rank of the algebra:the maximal number of mutually commuting generators.Crucial characteristics of nonabelian algebras.Rank determines the number of independent quantum numbers,which uniquely characterize each state within a given irreducible representation,or,as is common to say,multiplet.

2.3SU(2)group and algebra

The simplest of nonabelian Lie groups,with plenty of applications in particle physics is the SU(2)group. Moreover,most of the techniques useful for the more complicated case of SU(3)and other groups can be generalized from the technically simpler case of SU(2).

De?nition2.1SU(2)is formed by unitary2×2matrices with unit determinant.The associated Lie algebra is made out of traceless hermitian matrices2×2.

There are3independent generators J1,J2,J3of SU(2),which form the basis of SU(2)albegra and which satisfy the well-known commutation relations

[J i,J j]=iεijk J k.(2.20) Remark2.1In fact this algebra is equivalent to that of SO(3)group,indicating that the relation between the group and the associated algebra is not unique.This has to do with the fact that albegras express only the local properties of the groups,but don’t describe the global ones.

Representations of SU(2):can be constructed by means of various techniques.The one which proved to be the most useful for generalization to SU(3)is called the method of the highest weight.We shall be interested in the construction of irreducible representations only.

Construction:proceeds in several steps:

1.SU(2)is of rank1,i.e.there is only one operator,which fully characterizes the state within a given

multiplet.Let us choose J3for that purpose and denote the state with the highest weight j as|j :

J3|j =j|j ; i|j =δij.(2.21)

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