Gauge Theories in Particle Physics : QCD and The Electroweak Theory, Fourth Edition

512 Pages·2012·4.59 MB·English

by  Aitchison

Particle Physics Aitchison
Hey
FOURTH EDITION
FOURTH EDITION • VOLUME 2
GAUGE THEORIES

GAUGE THEORIES PARTICLE PHYSICS
IN
FOURTH
A PRACTICAL INTRODUCTION EDITION
I N

PARTICLE PHYSICS
“Aitchison and Hey was the ‘bible’ for me as a young post-doc in the 1980s … G
The book has been revised regularly as the field progressed and I am delighted A
to see a new edition which brings it up to date to the discovery of a Higgs-like U
A P R A C T I C A L I N T R O D U C T I O N
boson at the LHC in July 2012. … Its strength has always been the combination G
A
of the theory with discussion of experimental results. The new edition continues E

this tradition by including … discussion of the related important observations of P T VOLUME 2
the last ten years—CP violation and oscillations in the B sector and the now rich R
H
phenomenology of neutrino oscillations. This will become a new classic.” A
E Non-Abelian Gauge Theories
—Amanda Cooper-Sarkar, Oxford University, UK CO
TR QCD and The Electroweak Theory
“This is an indispensible textbook for all particle physicists, experimentalists and
I
theorists alike, providing an accessible exposé of the Standard Model, covering CIE
the mathematics used to describe it and some of the most important experi- AS
mental results which vindicate it. … Volume 2 has been updated with extended L
I
discussions on quark and neutrino mixing and inclusion of results on CP N
IP
violation and neutrino oscillations … these textbooks will remain on the top of a NA
high energy physicist’s reading list for years to come.” T
R
—Matthew Wing, University College London, UK R
T
O
New to the Fourth Edition I
DC
• New chapter on CP violation and oscillations in mesonic and neutrino systems UL
E
• New section on three-generation quark mixing and the CKM matrix C
TP
• Improved discussion of two-jet cross section in electron-positron annihilation
IH
• New section on jet algorithms O
Y
• Recent lattice QCD calculations with dynamical fermions N
S
• New section on effective Lagrangians for spontaneously broken chiral
I
symmetry, including the three-flavor extension, meson mass relations, and C
chiral perturbation theory S
• Update of asymptotic freedom

• Discussion of the historic discovery of a Higgs-like boson
VOLUME 2
Ian J.R. Aitchison • Anthony J.G. Hey
K14936
FOURTH EDITION
GAUGE THEORIES
I N

PARTICLE PHYSICS
A P R A C T I C A L I N T R O D U C T I O N
VOLUME 2
Non-Abelian Gauge Theories
QCD and The Electroweak Theory
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
FOURTH EDITION
GAUGE THEORIES
I N

PARTICLE PHYSICS
A P R A C T I C A L I N T R O D U C T I O N
VOLUME 2
Non-Abelian Gauge Theories
QCD and The Electroweak Theory
Ian J.R. Aitchison • Anthony J.G. Hey
Boca Raton London New York
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Contents
Preface xiii
V Non-Abelian Symmetries 1
12 Global Non-Abelian Symmetries 3
12.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . 3
12.2 The flavour symmetry SU(2) . . . . . . . . . . . . . . . . . . 5
f
12.2.1 The nucleon isospin doublet and the group SU(2) . . . 5
12.2.2 Larger (higher-dimensional) multiplets of SU(2) in nu-
clear physics . . . . . . . . . . . . . . . . . . . . . . . 12
12.2.3 Isospin in particle physics: flavour SU(2) . . . . . . . 14
f
12.3 Flavour SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . 18
f
12.4 Non-Abelian global symmetries in Lagrangian quantum field
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
12.4.1 SU(2) and SU(3) . . . . . . . . . . . . . . . . . . . . 24
f f
12.4.2 Chiral symmetry . . . . . . . . . . . . . . . . . . . . . 31
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
13 Local Non-Abelian (Gauge) Symmetries 39
13.1 Local SU(2) symmetry . . . . . . . . . . . . . . . . . . . . . 40
13.1.1 The covariant derivative and interactions with matter 40
13.1.2 The non-Abelian field strength tensor . . . . . . . . . 48
13.2 Local SU(3) Symmetry . . . . . . . . . . . . . . . . . . . . . 49
13.3 Local non-Abelian symmetries in Lagrangian quantum field
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
13.3.1 Local SU(2) and SU(3) Lagrangians . . . . . . . . . . 51
13.3.2 Gauge field self-interactions . . . . . . . . . . . . . . 54
13.3.3 Quantizing non-Abelian gauge fields . . . . . . . . . . 60
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
VI QCD and the Renormalization Group 71
14 QCD I: Introduction, Tree Graph Predictions, and Jets 73
14.1 The colour degree of freedom . . . . . . . . . . . . . . . . . . 74
14.2 The dynamics of colour . . . . . . . . . . . . . . . . . . . . . 78
14.2.1 Colour as an SU(3) group. . . . . . . . . . . . . . . . 78
14.2.2 Global SU(3) invariance, and ‘scalar gluons’ . . . . . 80
c
vii
viii
14.2.3 Local SU(3) invariance: the QCD Lagrangian . . . . 82
c
14.2.4 The θ-term . . . . . . . . . . . . . . . . . . . . . . . . 84
14.3 Hard scattering processes, QCD tree graphs, and jets . . . . 86
14.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 86
14.3.2 Two-jet events in p¯p collisions . . . . . . . . . . . . . 88
14.3.3 Three-jet events in p¯p collisions . . . . . . . . . . . . . 95
14.4 3-jet events in e+e− annihilation . . . . . . . . . . . . . . . . 97
14.4.1 Calculation of the parton-level cross section . . . . . . 98
14.4.2 Soft and collinear divergences . . . . . . . . . . . . . . 101
14.5 Definition of the two-jet cross section in e+e− annihilation . 103
14.6 Further developments . . . . . . . . . . . . . . . . . . . . . . 106
14.6.1 Test of non-Abelian nature of QCD in e+e− → 4 jets . 106
14.6.2 Jet algorithms . . . . . . . . . . . . . . . . . . . . . . 107
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
15 QCD II: Asymptotic Freedom, the Renormalization Group,
and Scaling Violations 113
15.1 Higher-order QCD corrections to σ(e+e− → hadrons): large
logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
15.2 The renormalization group and related ideas in QED . . . . 116
15.2.1 Where do the large logs come from? . . . . . . . . . . 116
15.2.2 Changing the renormalization scale . . . . . . . . . . 118
15.2.3 The RGE and large −q2 behaviour in QED . . . . . . 121
15.3 Back to QCD: asymptotic freedom . . . . . . . . . . . . . . . 124
15.3.1 One loop calculation . . . . . . . . . . . . . . . . . . . 124
15.3.2 Higher-order calculations, and experimental comparison 127
15.4 σ(e+e− → hadrons) revisited . . . . . . . . . . . . . . . . . . 128
15.5 A more general form of the RGE: anomalous dimensions and
running masses . . . . . . . . . . . . . . . . . . . . . . . . . . 130
15.6 QCD corrections to the parton model predictions for deep in-
elastic scattering: scaling violations . . . . . . . . . . . . . . 135
15.6.1 Uncancelled mass singularities at order α . . . . . . . 136
s
15.6.2 Factorization, and the order α DGLAP equation . . 142
s
15.6.3 Comparison with experiment . . . . . . . . . . . . . . 145
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
16 Lattice Field Theory, and the Renormalization Group Revis-
ited 151
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 152
16.2.1 Scalar fields . . . . . . . . . . . . . . . . . . . . . . . . 152
16.2.2 Dirac fields . . . . . . . . . . . . . . . . . . . . . . . . 154
16.2.3 Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . 158
16.3 Representation of quantum amplitudes . . . . . . . . . . . . 161
16.3.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . 162
ix
16.3.2 Quantum field theory . . . . . . . . . . . . . . . . . . 167
16.3.3 Connection with statistical mechanics . . . . . . . . . 171
16.4 Renormalization, and the renormalizationgroup, on the lattice 172
16.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 172
16.4.2 Two one-dimensional examples . . . . . . . . . . . . . 174
16.4.3 Connections with particle physics . . . . . . . . . . . . 177
16.5 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . 182
16.5.1 Introduction, and the continuum limit . . . . . . . . . 182
16.5.2 The static qq¯ potential. . . . . . . . . . . . . . . . . . 184
16.5.3 Calculation of α(M2) . . . . . . . . . . . . . . . . . . 187
Z
16.5.4 Hadron masses . . . . . . . . . . . . . . . . . . . . . . 189
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
VII Spontaneously Broken Symmetry 193
17 Spontaneously Broken Global Symmetry 195
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
17.2 The Fabri–Picasso theorem . . . . . . . . . . . . . . . . . . . 197
17.3 Spontaneously broken symmetry in condensed matter physics 199
17.3.1 The ferromagnet . . . . . . . . . . . . . . . . . . . . . 199
17.3.2 The Bogoliubov superfluid . . . . . . . . . . . . . . . . 202
17.4 Goldstone’s theorem . . . . . . . . . . . . . . . . . . . . . . . 209
17.5 Spontaneously broken global U(1) symmetry: the Goldstone
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
17.6 Spontaneously broken global non-Abelian symmetry . . . . . 216
17.7 The BCS superconducting ground state . . . . . . . . . . . . 219
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
18 Chiral Symmetry Breaking 227
18.1 The Nambu analogy . . . . . . . . . . . . . . . . . . . . . . . 228
18.1.1 Two flavour QCD and SU(2) ×SU(2) . . . . . . . 231
fL fR
18.2 Pion decay and the Goldberger–Treimanrelation . . . . . . . 235
18.3 Effective Lagrangians . . . . . . . . . . . . . . . . . . . . . . 239
18.3.1 The linear and non-linear σ-models . . . . . . . . . . . 239
18.3.2 Inclusion of explicit symmetry breaking: masses for pi-
ons and quarks . . . . . . . . . . . . . . . . . . . . . . 245
18.3.3 Extension to SU(3) ×SU(3) . . . . . . . . . . . . . 247
fL fR
18.4 Chiral anomalies . . . . . . . . . . . . . . . . . . . . . . . . . 249
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
19 Spontaneously Broken Local Symmetry 255
19.1 Massive and massless vector particles . . . . . . . . . . . . . 255
19.2 Thegenerationof‘photonmass’inasuperconductor: Ginzburg–
Landau theory and the Meissner effect . . . . . . . . . . . . 260
19.3 SpontaneouslybrokenlocalU(1)symmetry: theAbelianHiggs
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264