Gauge Theories in Particle Physics, Volume II: A Practical Introduction : Non-Abelian Gauge Theories : Qcd and the Electoweak Theory (Graduate Student Series in Physics)

450 Pages·2003·4.46 MB·English

by  I.J.R. Aitchison, A.J.G. Hey

GRADUATE STUDENT SERIES IN PHYSICS

Series Editor:
Professor Douglas F Brewer, MA, DPhil
Emeritus Professor of Experimental Physics, University of Sussex

GAUGE THEORIES IN
PARTICLE PHYSICS

A PRACTICAL INTRODUCTION
THIRD EDITION

Volume 2

Non-Abelian Gauge Theories:
QCD and the Electroweak Theory

IAN J R AITCHISON
Department of Physics
University of Oxford

ANTHONY J G HEY
Department of Electronics and Computer Science
University of Southampton

INSTITUTE OF PHYSICS PUBLISHING
Bristol and Philadelphia

c

� IOP Publishing Ltd 2004

All rights reserved.
No part of this publication may be reproduced, stored
in a retrieval system or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the prior permission
of the publisher. Multiple copying is permitted in accordance with the terms
of licences issued by the Copyright Licensing Agency under the terms of its
agreement with Universities UK (UUK).

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

ISBN 0 7503 0950 4

Library of Congress Cataloging-in-Publication Data are available

Front cover image: Simulation by the ATLAS experiment of the decay of a
Higgs boson into four muons (yellow tracks). c

� CERN Geneva.

Commissioning Editor: John Navas
Production Editor: Simon Laurenson
Production Control: Sarah Plenty and Leah Fielding
Cover Design: Victoria Le Billon
Marketing: Nicola Newey and Verity Cooke

Published by Institute of Physics Publishing, wholly owned by The Institute of
Physics, London

Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK
US Office: Institute of Physics Publishing, The Public Ledger Building, Suite
929, 150 South Independence Mall West, Philadelphia, PA 19106, USA

Typeset in LATEX2

� by Text 2 Text Limited, Torquay, Devon
Printed in the UK by MPG Books Ltd, Bodmin, Cornwall

To Jessie
and to
Jean, Katherine and Elizabeth

CONTENTS

P ref a c e t o Vo lume 2 o f t he Third Edit io n

Acknowledgments

PA R T 5
N O N - A BELIA N SY MMETR IES

12 Global no n-Abelian Symmetries

12.1 The flavour symmetry SU(2)f

12.1.1 The nucleon isospin doublet and the group SU(2)
1 2 . 1 . 2 L arg e r ( h ig h e r- d im e n sio n a l) m u ltip lets o f SU( 2 ) in
nuclear physics

12.1.3 Isospin in particle physics

12.2 Flavour SU(3)f
12.3 Non-Abelian global symmetries in Lagrangian quantum field
th eo r y
12.3.1 SU(2)f and SU(3)f
12.3.2 Chiral symmetry
Problems

1 3 Lo ca l no n- A belia n ( G a ug e) Sy mmet r ies

13.1 Local SU(2) symmetry: th e covariant derivative and interactions
with m a tter

13.2 Covariant derivatives and coordinate transformations
13.3 Geometrical curvature and th e g auge field strength tenso r
13.4 Local SU(3) symmetry
13.5 Local non-Ab elian symmetries in Lagrangian quantum field th eory

13.5.1 Local SU(2) and SU(3) Lagrangians
13.5.2 Gauge field self-interactions
13.5.3 Quantizin g non-Ab elian g auge field s
Problems

PA R T 6
Q C D A N D TH E R EN O R MA LIZATIO N G RO U P

14 QCD I: Intro duction and Tree-Graph Predictio ns

14.1 The colour degree of freedom
14.2 The dynamics of colour

14.2.1 Colour as an SU(3) g roup
14.2.2 Global SU(3)c invariance and ‘scalar gluons’
14.2.3 Local SU(3)c invariance: the QCD Lagrangian

14.3 Hard scattering processes and QCD tree g raphs
14.3.1 Two -jet events in ¯pp co llisio n s
14.3.2 Three-jet events

14.4 Three-jet events in e+ e− annihilation

Problems

15 QCD II: Asymptotic Freedom, the Renormalization Group and
Sca ling Vio la t io ns in Deep Inelastic Scattering
15.1 QCD corrections to th e p arto n model p redictio n for σ(e+ e− →
hadrons)

15.2 The renormalizatio n g roup and related id eas

15.2.1 Where do th e large logs come from?
15.2.2 Changing th e renormalizatio n scale
15.2.3 The renormalization group equation and large − q 2

behaviour in QED

15.3 Back to QCD: asymptotic freedom
15.4 A more general form of the RGE: anomalous dimensions and
running masses

1 5 . 5 So m e tech n icalities
15.6 σ(e+ e− → hadrons) revisited
15.7 QCD corrections to th e p arto n model p redictions for d eep
in elastic scatter in g : scalin g v io latio n s
15.7.1 Uncancelled m ass singularities
15.7.2 Factorizatio n and th e DGLAP equatio n
15.7.3 Compariso n with experiment
Problems

16 La ttice Field Theory a nd the Renorma lizatio n Gro up Revisited

16.1 Introduction
1 6 . 2 Discr etizatio n
1 6 . 3 Gau g e inva r ian ce o n th e lattice
16.4 Representatio n o f quantum amplitudes
1 6 . 5 Co n n ectio n with statistical m ech an ics
1 6 . 6 Ren o r m a lizatio n a n d th e r en o r m a lizatio n g r o u p o n th e lattice

16.6.1 Introduction
16.6.2 The one-dimensional Ising model

16.6.3 Further developments and some connections with
particle physics

16.7 Numerical calculations

Problems

PA R T 7
SP O N TA N EO U SLY BRO K EN S Y MMETRY

17 Spontaneously Broken Global Sy mmetry

17.1 Introduction
17.2 The Fabri– Picasso th eorem
17.3 Spontaneously broken sy mmetry in condensed matter physics
17.3.1 The ferromagnet
17.3.2 The Bogoliubov superfluid

17.4 Goldstone’s theorem
17.5 Spontaneously broken global U(1) symmetry: the Goldstone
model

17.6 Spontaneously broken global non-Abelian symmetry
17.7 The BCS superconductin g g round state

Problems

18 Chiral Symmetry Breaking

18.1 The Nambu analogy

18.1.1 Two flavour QCD and SU(2)f L×SU( 2 ) f R

18.2 Pion d ecay and the Goldberg er–Treiman relation
18.3 The lin ear and nonlin ear σ -models
18.4 Chiral anomalies

Problems

1 9 Spo nt a neo usly Bro ken Lo ca l Sy mmet r y

19.1 Massive and m assless vector particles
19.2 The generation of ‘photon mass’ in a superconducto r: th e
Meissn er effect

19.3 Spontaneously broken local U(1) symmetry: the Abelian Higgs
model

19.4 Flu x quantizatio n in a superconducto r
19.5 ’t Hooft’s gauges
19.6 Spontaneously broken lo cal SU(2)×U(1) symmetry

Problems

PA R T 8
WEA K IN TER AC TIO N S A N D TH E ELEC TROW EA K TH EO RY

20 Introduction to the Phenomeno logy of Weak Interactio ns

20.1 Fermi’s ‘current–current’ theory o f nuclear β -decay and its
generalizations

20.2 Parity violatio n in weak in teractions
20.3 Parity transformatio n o f Dirac wavefunctions and field operato rs
20.4 V − A th e o r y : ch ir ality an d h elicity
20.5 Charg e conjugatio n for fermio n wavefunctions and field operato rs
20.6 Lepton number
20.7 The universal current–current th eory for weak in teractions of
leptons

20.8 Calculation o f the cross-section for νµ + e− → µ− + νe
20.9 Leptonic weak neutral currents
20.10 Qu ark weak currents
20.11 Deep in elastic neutrino scattering
20.12 Non-leptonic weak in teractions

Problems

2 1 Dif ficult ies wit h t he Current – Current a nd ‘Naive’ Intermediate
Vector Bo so n Models
21.1 Vio latio n o f unitarity in th e current–current model
21.2 The IVB model
21.3 Violation of unitarity bounds in the IVB model
2 1 . 4 T h e p r o b lem o f n o n - r e n o r m a lizab ility in weak in ter actio n s

Problems

22 The Glashow–Salam–Weinberg Gauge Theory of Electroweak
Interactions
22.1 Weak isospin and hypercharg e: th e SU(2) × U(1) group of the
electroweak in teractions: quantum number assignments and W
a n d Z m a sse s

22.2 The leptonic currents (massless neutrinos): relation to current–
current model

22.3 The quark currents
22.4 Simple (tree-level) predictions
22.5 The discovery of the W± and Z0 at the CERN p¯p c o llid er

22.5.1 Productio n cross-sections for W and Z in p¯p c o llid er s
22.5.2 Charg e asy mmetry in W± decay
22.5.3 Discovery of the W± and Z0 at the p¯p c o llid er an d th e ir
properties

22.6 The fermio n mass p roblem
22.7 Three-family mixing

22.7.1 Quark flavour mixing

22.7.2 Neutrino flavour mixing

22.8 Higher-order corrections
22.9 The top quark
22.10 The Higgs sector

22.10.1 Introduction
22.10.2 Theoretical consid erations concerning mH
22.10.3 Higgs phenomenology
Problems

Appendix M
G ro up Theo r y

M.1
Defin itio n a n d sim p le ex am p les

M.2
Lie groups

M.3
Generators o f Lie groups

M.4
Examples
M.4.1 SO(3) and three-dimensional rotations
M.4.2 SU(2)
M.4.3 SO(4): The sp ecial orthogonal g roup in four dimensions
M.4.4 The Lorentz group
M.4.5 SU(3)

M.5
Matrix representations of generato rs and o f Lie groups

M.6
The Lorentz group

M.7
The relatio n b etween SU(2) and SO(3)

Appendix N
Dimensio na l R eg ula r iza t io n

Appendix O
G r a ssma nn Va r ia bles

Appendix P
Ma j o ra na Fermio ns

P.1
Spin- 1

2 wave equations

P.2
Majorana quantum field s

Appendix Q
Fe y nma n R ules fo r Tre e G ra phs in Q C D a nd t he
Electroweak Theory
Q.1
QCD
Q.1.1 External p articles
Q.1.2 Propagato rs
Q.1.3 Vertices

Q.2
The electroweak theory
Q.2.1 External p articles
Q.2.2 Propagato rs
Q.2.3 Vertices

References

PREFACE TO VOLUME 2 OF THE THIRD EDITION

Volume 1 of our new two-volume third edition covers relativistic quantum
mechanics, electromagnetism as a gauge theory, and introductory quantum
field theory, and leads up to the formulation and application of quantum
electrodynamics (QED), including renormalization.
This second volume is
devoted to the remaining two parts of the ‘Standard Model’ of particle physics,
namely quantum chromodynamics(QCD) and the electroweak theory of Glashow,
Salam and Weinberg.
It is remarkable that all three parts of the Standard Model are quantum
gauge field theories:
in fact, QCD and the electroweak theory are certain
generalizations of QED. We shall therefore be able to build on the foundations of
gauge theory, Feynman graphs and renormalization which were laid in Volume 1.
However, QCD and the electroweak theory both require substantial extensions
of the theoretical framework developed for QED. Most fundamentally, the
discussion of global and local symmetries must be enlarged to include non-
Abelian symmetries, and spontaneous symmetry breaking. At a somewhat more
technical level, the lattice (or path-integral)approach to quantum field theory, and
the renormalization group are both needed for access to modern work on QCD.
For each of these theoretical elements, a self-contained introduction is provided
in this volume. Together with their applications, this leads to a simple four-part
structure (the numbering of parts, chapters and appendices continues on from
Volume 1):

Par t 5 No n - Ab elian sy m m etr ies
Pa r t 6 QCD an d th e r e n o r m a lizatio n g r o u p ( in c lu d in g lattice field th e o r y )
Part 7 Spontaneous symmetry breaking (including the spontaneous breaking of
the approximate global chiral symmetry of QCD)
Part 8 The electroweak theory.

We have already mentioned several topics (path integrals, the renormaliza-
tion group, and chiral symmetry breaking) which are normally found only in texts
pitched at a more advanced level than this one—and which were indeed largely
omitted from the preceding (second) edition. Nor, as we shall see, are these topics
the only newcomers. With their inclusion in this volume, our book now becomes a
comprehensive, practical and accessible introduction to the major theoretical and
experimental aspects of the Standard Model. The emphasis is crucial: in once
again substantially extending the scope of the book, we have tried hard not to

compromise the title’s fundamental aim—which is, as before, to m ake the chosen
material accessib le to th e wid e read ersh ip wh ich th e p r ev io u s ed itio n s ev id en tly
attracted.
A glance at the contents will suggest that we have set ourselves a
consid erable challenge. On the other h and, not all o f the topics are likely to b e
of equal interest to every reader. It may th erefore b e h elpful to offer some more
d etailed g u id an ce, wh ile at th e sam e tim e highlighting th o se item s wh ich a r e n ew
to th is ed itio n .
First, th en, non-Ab elian symmetry. This refers to th e fact th at th e symmetry
transf ormations are m atrices (acting o n a set o f fields), any two of which will
generally not commute with each other, so th at the o rder in which they are applied
makes a difference. Much of the n ecessary mathematics already appears in the
simpler case in which th e symmetry is a global, rath er th an a local one. In chapter
12 we introduce global non-Abelian symmetries via the physical examples of the
(approximate) SU(2) and SU(3) flavour symmetries of the strong interactions.
The underlying mathematics involved here is group theory. However, we
take care to d evelop everything we need on a ‘do-it-yourself ’ b asis as we go along,
so th at no prio r knowledge of group th eory is necessary. Neverth eless, we have
prov ided a n ew and fairly serious appendix (M) on group theory, which collects
togeth er th e main relevant ideas, and sh ows h ow th ey apply to the groups we are
dealing with (including the Lorentz group). We hope that this compact summary
will be of use to those readers who want a sense of the mathematical unity behind
the succession of sp ecific calculatio n s p r ov id ed in th e m ain tex t.
A further important global non-Abelian symmetry is also introduced in
ch ap ter 1 2 —th at o f ch ir al sy m m e tr y, wh ich is ex p ected to b e r e leva n t if th e q u a r k
masses are substantially less than typical hadronic scales, as is indeed the case.
The apparent non-observatio n o f this expected sy mmetry creates a puzzle, th e
reso lu tio n o f which has to b e d eferred until part 7.

In chapter 13, the second in part 5, we move on to the local versions of
SU(2) and SU(3) symmetry, arriving in section 13.5 at the corresponding non-
Ab elian g auge field th eories wh ich are th e main focus of th e book, bein g d irectly
relevant to th e electroweak th eory and to QCD resp ectively. Crucial n ew physical
phenomena appear, not present in QED—for ex ample, th e self-interactions among
the gauge field quanta.
On th e mathematical side, the algebraic (or group-th eoretic) asp ects
developed in chapter 12 carry th rough unchanged into chapter 13, but the
‘gauging’ of the symmetry brings in some new geometrical concepts, such as
‘covariant derivative’, ‘parallel transport’, ‘connection’, and ‘curvature’. We
decided again st banish in g these matters to an appendix, since they are su ch a
significant p art o f the conceptual structure o f all gauge th eories, and moreover
th eir in c lu sio n allows in str u ctive r ef er en ce to b e m a d e to a th e o r y o th er wise
ex cluded from m ention, namely general r elativ ity. All the same, practically-
minded readers may want to pass quickly over sections 13.2 and 13.3, and also
section 13.5.3, which explains why obtaining the correct Feynman rules for loops

in a non-Ab elian g auge th eory is su ch a d ifficult p roblem, with in th e ‘canonical’
approach to quantum field theory as developed in volume 1.
I m m e d iate a p p licatio n o f th e f o r m alism can n ow b e m ad e to QCD, a n d th is
occupies most of th e n ex t three chapters, which form part 6. Ch apter 1 4 introduces
‘colour’ as a dynamical degree of freedom, and leads on to the QCD Lagrangian.
Some simple tree-graph applications ar e then d escrib ed, u sing th e techniques
learned for QED. These p rovide a good first orientatio n to d ata, following ‘parto n
model’ and ‘ scaling’ ideas.
Bu t o f course a fundamental questio n immediately arises: how can su ch
an approach, b ased on perturbatio n theory, possibly apply to QCD wh ich, after
all, describes the strong in teractions between quarks?
The answer lies in
the profound property (possessed only by non-Abelian gauge theories) called
‘asy mptotic freedom’—that is, the decrease o f the effective interaction strength
at high energies or short d istances. Crucially, this p roperty cannot be understood
in terms of tree graphs: loops must be studied, and this immediately involves
renormalization. In fact, p ertu rbatio n theory b ecomes u seful at h ig h energ ies only
after an infinite series of loop contribu tions has b een effectively re-su mmed. The
technique required to do this goes by the name of th e ren o rma liza tio n g ro u p (RG),
and it is d escribed in chapter 15, along w ith applications to asymptotic freedom,
an d to th e calcu latio n o f scalin g v io latio n s in d eep in elastic scatter in g .
We do not expect the majority of our readers to find chapter 15 easy going.
Bu t there is no deny in g the central importance o f the RG in modern field th eory,
nor its direct relevance to experiment. In sectio n 15.2 we h ave tried to prov id e
an elementary in troductio n to the RG, b y consid erin g in d etail the much simpler
case o f QED, u sing no more th eory th an is contained in chapter 11 of vo lu me 1.
Sections 15.4 and 15.5 are less central to the m ain argument, as is an appendix
(N) on dimensional regularization.
In chapter 16, the third of part 6, we turn to the problem of how to
extract predictions from a quantum field theory (in particular, QCD) in the non-
perturbative regime. The available technique is computational, based on the
discretized (lattice) version o f Feynman’s path-integral formulation of quantum
fi e ld th eo ry , to which we provide a simple introduction in section 16.4. A
substantial bonus of this formulation is that it allows fruitful analogies to be
d r awn with th e statistical m ech an ics o f sp in sy stem s o n a lattice. I n p ar ticu lar,
we hope that readers who may have struggled with the formal manipulations of
chapter 1 5 will be refresh ed by seeing RG ideas in action from a diff erent and
more physical point of view—that of ‘integrating out’ short distance degrees of
freedom, leading to an effective theory valid at longer d istances. The chapter ends
with so m e illu str a tive r e su lts f r o m lattice QCD calcu latio n s, in sectio n 1 6 . 7 . An
appendix (O) on Grassmann variables is provided for those interested in seeing
how the path-integral formalism can be made to work for fermions.
At this half-way stage, QCD has been established as the theory of strong
interactions, by the success of both RG-improved perturbation theory and non-
perturbative numerical computations.

Fu rther p rogress requires one more fundamental id ea—th e subtle concept o f
spontaneous symmetry breaking, which forms the subject of part 7. Chapter 17
sets out the basic theory of spontaneously broken global symmetries, and also
consid ers two physical ex amples in consid erable detail, namely the Bogoliubov
superfluid in sectio n 1 7 . 3 , an d the BCS superconductor in sectio n 1 7 . 7 . I t is
of course true that these systems are not part of the standard model of particle
physics. However, th e characteristic methods and concepts developed for su ch
systems provide valuable background for the particle physics applications of the
id ea, wh ich follow in the next two chapters. In particular, our presentatio n o f
ch ira l symmetry b re a k in g in chapter 18 follows Nambu’s remarkable original
analogy between fermion m ass g eneratio n and the appearance of an energy gap in
a superconductor. Section 18.3, on lin ea r a n d n o n lin ea r sig ma mo d e ls, is rather
more optional, as is our brief introduction to chiral anomalies in sectio n 1 8 . 4 .
In chapter 19, the third in part 7, we consider the spontaneous breaking of
local (gauge) symmetries. Here the fundamental point is that it is possible for
g a u g e q u a n ta to acq u ir e m a ss, wh ile still p r eser v in g th e lo cal g a u g e sy m m e tr y
of th e Lagrangian. We consid er applications both to the Ab elian case o f a
superconductor (sections 19.2 and 19.4—once again, a valuable working model
of th e physics), and to th e non-Ab elian case required for th e electroweak th eory.
The way is now clear to develop the electroweak theory, in p art 8 . Chapter 20

is a self-contained rev iew o f weak in teractio n phenomenology, b ased on Fermi’s
‘current–current’ model. New material h ere includes d iscussion of th e d iscrete
symmetries C and P , and of lepton number conservation taking into account the
p o ssib ility th at neutrinos may be Majorana particles, in support of which we
provide an appendix (P) on Majorana fermions. Chapter 21 describes what goes
wrong with the current–current model, and with theories in which the W and Z
bosons are g iven a ‘naive’ mass, and suggests why a g auge th eory is needed to
avo id th e se d ifficu lties. Fin a lly, in c h a p ter 2 2 , all th e p ieces are p u t to g e th er in
the presentation of the electroweak theory. New additions here include three-
family mixing via the CKM matrix, together with more detail on higher order
(one-loop) corrections, the top quark, and aspects of Higgs phenomenology. The
remarkably precise agreement—thus far—between theory and experiment, which
depends upon the inclusion of one-loop effects, makes it hard to deny that, when
interacting weakly, Nature has indeed made use of the subtle intricacies of a
renormalizable, spontaneously broken, non-Abelian chiral gauge theory.
But the story of the Standard Model is not yet quite complete.
One
vital part—the Higgs sector—remains virtual, and phenomenological. Further
progress in understanding the mechanism of electroweak symmetry breaking,
and of mass generation, requires input from the next generation of experiments,
primarily at the LHC. We hope that we leave our readers with a sound grasp of
what is at stake in these experiments, and a lively interest in their outcome.

Acknowledg ments

Our expression of thanks to friends and colleagues, made in the first volume,
ap p lies e q u a lly to th is o n e . M o r e p ar ticu lar ly : Keith Ham ilto n was a willin g
‘guinea pig’ for chapters 14–17, and we thank him for his careful reading and
en co u r ag in g co m m en ts; we ar e g r atef u l to Ch r is Allto n f o r ad v ice o n lattice g au g e
th eo r y r e su lts, f o r sectio n 1 6 . 7 ; an d Nik k i Fa th er s a g a in p r ov id ed essen tial h elp in
making the electronic version. Above all, the constant and unstinting help of our
good friend George Emmons throughout the genesis and production of the book
h a s, o n c e a g a in , b een inva lu ab le.
In addition, we are delighted to thank two new correspondents, whom we
look forward to g reetin g physically, as well as electronically. Paolo Strolin and
Peter Williams both worked very carefully through Vo lume 1, and b etween them
found a good many misp rints and infelicities. An up-to-date list will be posted o n
the book’s website:
http://bookmarkphysics.iop.org/bookpge.htm?&book=1130p.
Paolo also r ead drafts of chapters 12 an d 14, and made many u seful comments.
We wish it had b een possible to send h im more chapters: h owever, h e m ade
numerous excellent suggestions for improving our treatment of weak interactions
in th e seco n d ed itio n ( p ar ts 4 an d 5 ) , an d —wh er e f easib le—m any o f th em h ave
b een in co r p o r ated in to th e p r e sen t p a r t 8 . Er r o r s, o ld an d n ew, th e r e still will b e ,
of course: we hope readers will draw them to our attention.

Ian J R Aitchison and Anthony J G Hey
October 2003

PART 5

NON-ABELIAN SYMMETRIES

12

GLOBAL NON-ABELIAN SYMMETRIES

In the p receding volume, a very successful dynamical theory—QED—has
been introduced, b ased on the remarkably simple gauge principle: n amely
that the theory should be invariant under local phase transformations on the
wavefunctions (chapter 3) or field operato rs (chapter 7) of charged p articles.
Such transf ormations were characterized as Ab elia n in sectio n 3 .6 , sin ce th e
phase factors commuted. The second volume of this book will be largely
co n cer n e d with th e f o r m u latio n a n d elem en tar y ap p licatio n o f th e r e m a in in g two
dynamical th eories with in th e Standard Model—th at is, QCD and the electroweak
th eory. They are bu ilt on a g eneralizatio n o f the gauge prin ciple, in wh ich the
tr an sf o r m a tio n s invo lve m o r e th a n o n e state, o r field , a t a tim e. I n th at case,
the ‘ phase factors’ become matrices, which generally do not commute with each
other, and the asso ciated sy mmetry is called a ‘non-Abelian ’ one. When the phase
factors are independent of th e spacetime coordinate x , the symmetry is a ‘global
non-Abelian’ one; when they are allowed to depend on x , one is led to a non-
Abelian gauge theory. Both QCD and the electroweak theory are o f the latter
type, providing generalizations of the Abelian U(1) gauge theory which is QED.
It is a strik in g fact th at all three dynamical th eories in th e Standard Model are
based o n a gauge prin ciple o f local phase invariance.
In th is chapter we shall be main ly concerned with two g lobal non-Ab elian
sy m m e tr ies, wh ich lead to u sef u l co n ser va tio n laws bu t n o t to any sp ecific
dynamical th eory. We b eg in in sectio n 12.1 with th e first non-Ab elian symmetry
to be used in particle physics, the hadronic isospin ‘SU(2) symmetry’ proposed
by Heisenberg (1932) in the context of nuclear physics, and now seen as following
f r o m th e n ear eq u a lity o f th e u an d d q u a r k m a sses ( o n ty p ical h a d r o n ic scales) ,
and the flavour independence of the QCD interquark forces. In section 12.2 we
extend this to SU(3)f flavour symmetry, as was first done by Gell-Mann (1961)
and Ne’eman (1961)—an extension seen, in its turn, as reflecting the rough
equality of the u, d and s quark masses, together with flavour independence of
QCD. The ‘wavefunction’ approach of sections 12.1 and 12.2 is then reformulated
in field-theoretic language in section 12.3.

In the last section of this chapter, we shall introduce the idea of a global
chiral symmetry, which is a symmetry of theories with massless fermions. This
may be expected to be a good approximate symmetry for the u and d quarks. But
the anticipated observable consequences of this symmetry (for example, nucleon