Table Of ContentGauge theory of
elementary particle
physics
Problems and solutions
TA-PEI CHENG
University of Missouri - St. Louis
and
LING-FONG LI
Carnegie Mellon University
CLARENDONPRESS . OXFORD
2000
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Preface
Students of particle physics often find it difficult to locate resources to learn
calculational techniques. Intermediate steps are not usually given in the research
literature. To a certain extent, this is also the case even in some of the textbooks.
In this book of worked problems we have made an effort to provide enough details
so that a student starting in the field will understand the solution in each case.
Our hope is that with this step-by-step guidance, students (after first attempting
the solution themselves) can develop their skill, and confidence in their ability, to
work out particle theory problems.
This collection of problems has evolved from the supplemental material devel-
oped for a graduate course that one of us ( L.F.L.) has taught over the years at
Carnegie Mellon University, and is meant to be a companion volume to our text-
book Gauge Theory of Elementary Particle Physics (referred to as CL throughout
this book) rather than a complete assemblage of gauge theory problems. Neverthe-
less, it has a self-contained format so that even a reader not familiar with CL can
use it effectively. All the problems (usually with several parts) have been given a
descriptive title. By simply inspecting the table of contents readers should be able
to pick out the areas they wish to tackle.
Several new subjects have entered in the field in the fifteen years since the
original writing of CL. Although we have not revised the book to incorporate them
because we would not be able to do them justice, we hope this set of problem/
solution presentations is the first step towards remedying the situation. We have
incorporated a number of new topics and developed further those that were only
introduced briefly in the original text. Listed below are some of these areas:
o Relations among different renormalization schemes
r
Further applications of the path-integral formalism
o General relativity as a gauge theory
r
Superconductivity as a Higgs phenomenon
o Non-linear sigma model and chiral symmetry
o Path integral derivation of the axial anomaiy
o Infrared and collinear divergence in QCD
o Further examples of the parton model phenomenology
o QCD and A1 - j rule in the non-leptonic weak decays
o More on gauge theories of lepton number violation
r
Group theory of grand unification
o Further examples of solitons
Many people have helped us in preparing this book. Our thanks go particularly
to all the students who have taken the course and have worked through a good part
of these problems. One of us (T.P.C.) also wishes to acknowledge the enjoyable
vi
Prefoce
hospitality of the Santa Cruz Institute of Particle Physics when finishing up this
project . The original literature has only been referenced casually, and we apologize
to the authors whose work we may have neglected to cite.
This book and CL share a page on the World Wide Web at the URL
ht.tp z / /vrvw. umsl . edu/-tpcheng/gaugebooks . html. Misprints or
other corrections brought to our attention will be posted on this page. We would
be grateful for any comments about these books.
St. Louis T.P.C.
Pittsburgh L.F.L.
Jantary 1999
Contents
1 Field quantization
1.1
Simple exercises in ),Qa theory
1.2 1
Auxiliary field
6
1.3
Disconnected diagrams 8
I.4
Simple external field problem 9
1.5
Path integral for a free particle 1l
1.6
Path integral for a general quadratic action 13
1.7
Spreading of a wave packet t6
1.8
Path integral for a harmonic oscillator 11
1.9
Path integral for a partition function 2t
1.10
Partition function for an SHO system 23
1.11
Non-standardpath-integralrepresentation 25
Llz
Weyl ordering of operators 26
l.13
Generating functional for a scalar field 32
l.I4
Poles in Green's function 35
2
Renormalization
2.1
Counterterms in )"Qa theory and in QED 37
2.2
Divergences in non-linear chiral theory 39
2.3
Divergences in lower-dimensional field theories 4l
2.4
n -Dimensional 'spherical' coordinates 43
2.5
Some integrals in dimensional regularization 46
2.6
Vacuum polarization and subtraction schemes 49
2.1
Renormalization of ),Q3 theory in n dimensions 53
2.8
Renormalization of composite operators 57
2.9
Cutkosky rules 59
3 Renormalization group
3.1
Homogeneousrenormalization-groupequation 63
3.2
Renormalizatron constants 64
3.3
B-function for QED 67
3.4
Behaviour of g near a simple fixed point 69
3.5
Running coupling near a general fixed point 70
3.6
One-loop renormalization-group equation in massless ,1,@a theory 7t
3.7
B-function for the Yukawa coupling 72
3.8
Solving the renormarization-group equation by coleman's method 75
3.9
Anomalous dimensions for composite operators 77
Contents
4 :;"", theory and the quark model
4.1 Unitary and hermitian matrices 78
4.2 SU(n) matrices 19
4.3 Reality of SU(2) representations 79
4.4 An identity for unitary matrices 81
4.5 An identity for SU(2) matrices 82
4.6 SU(3) algebra in terms of quark fields 83
4.7 Combining two spin-] states 85
4.8 The SU(2) adjoint representation 87
4.9 Couplings of SU(2) vector representations 89
4.r0 Isospin breaking effects 90
4.rl Spin wave function of three quarks 93
4.r2 Permutation symmetry in the spin-isospin space 96
4.13 Combining two fundamental representations 97
4.t4 SU(3) invariant octet baryon-meson couplings 100
4.15 Isospin wave functions of two pions 105
4.16 Isospins in non-leptonic weak processes r07
5 Chiral symmetry
5.1 Another derivation of Noether's current 1r0
5.2 Lagrangran with second derivatives 111
5.3 Conservation laws in a non-relativistic theory 113
5.4 Symmetries of the linear o-model 115
5.5 Spontaneous symmetry breaking in the o-model r22
5.6 PCAC in the o-model r23
l
5.7 Non-linear o-model 126
5.8 Non-linear o-model II t28
5.9 Non-linear o-model III 130
5.10 SSB by two scalars in the vector representation 133
6 Renormalization and symmetry
6.1 Path-integral derivation of axial anomaly r36
6.2 Axial anomaly and 11 --> yy 140
6.3 Soft symmetry breaking and renormalizabllity 142
6.4 Calculation of the one-loop effective potential 143
7 The Parton model and scaling
7.1 The Gottfried sum rule t46
7.2 Calculation of OPE Wilson coefficients r47
7.3 o,o,(e+e- --> hadrons) and short-distance physics 151
7.4 OPE of two charged weak currents 155
7.5 The total decay rate of the W-boson 156
8 Gauge symmetries
8.1 The gauge field in tensor notation 158
t6r
8.2 Gauge field and geometry
8.3 General relativity as a gauge theory r63
Contents IX
8.-1 O(n ) gauge theory 165
8.5 Broken generators and Goldstone bosons r67
8.6 Symmetry breaking by an adjoint scalar r69
8.7 Symmetry breaking and the coset space 17t
s.8 Scalar potential and first-order phase transition t72
s.9 Superconductivity as a Higgs phenomenon t73
9 Ouantum gauge theories
9.1 Propagator in the covariant R6 gauge 175
9.1 The propagator for a massive vector field 176
9.-1 Gauge boson propagator in the axial gauge 177
9.-1 Gauge boson propagator in the Coulomb gauge 178
9.-s Gauge invariance of a scattering amplitude 180
e.6 Ward identities in QED 180
9.t Nilpotent BRST charges 184
e.8 BRST charges and physical states 186
10 Ouantum chromodynamics
r0.I Colour factors in QCD loops 188
I (1.2 Running gauge coupling in two-loop t9l
r 0.-3 Cross-section for three-jet events 193
n 0.+ Operator-product expansion of two currents 198
I (1.5 Calculating Wilson coefficients 20r
11 Electroweak theory
I1.1 Chiral spinors and helicity states 205
l l.l The polartzation vector for a fermion 206
i r.3 The pion decay rate and f, 208
r l.-l Uniqueness of the standard model scalar potential 212
I 1.5 Electromagnetic and gauge couplings 213
l 1.6 Fermion mass-matrix diagonalization 214
I 1.7 An example of calculable mixing angles 2r5
r 1.8 Conservation of the B - L quantum number 2t6
12 Electroweak phenomenology
1t.l Atomic parity violation 218
11.2 Polarization asymmetry of Z -- f f 221
1-.J Simple z-lepton decays 222
n.-t Electron neutrino scatterings 223
11.5 CP properties of kaon non-leptonic decays 225
1t.6 Z --> H H is forbidden 226
j
t2.7 A,I - enhancement by short-distance QCD 227
n.8 Scalar interactions and the equivalence theorem 230
l].9 Two-body decays of a heavy Higgs boson 234
Contents
13 -0,"" in flavourdynamics
13. l Anomaly-free condition in a technicolour theory 238
13.2 Pseudo-Goldstone bosons in a technicolour model 239
13.3 Properties of Majorana fermions 239
13.4 LL --> ey and heavy neutrinos 244
13.5 Leptonic mixings in a vector-like theory 250
13.6 Muonium-antimuonium transition 252
14 Grand unification
t4.t Content of SU(5) representations 255
14.2 Higgs potential for SU(5) adjoint scalars 256
14.3 Massive gauge bosons in SU(5) 258
14.4 Baryon number non-conserving operators 260
14.5 SO(n) group algebra 260
14.6 Spinor representations of SO(n ) 263
14.7 Relation between SO(2n) and SU(n ) groups 267
14.8 Construction of SO(2n) spinors 269
15 Magnetic monopoles
15.1 The Sine-Gordon equation 275
15.2 Planar vortex field 280
15.3 Stability of soliton 282
15.4 Monopole and angular momentum 283
16 lnstantons
t6.t The saddle-point method 289
16.2 An application of the saddle-point method 292
t6.3 A Euclidean double-well problem 295
References 301
lndex 303
1
Field quantization
1 .1 Simple exercise s in ),Qa theory
In )"Qa theory, the interaction is given by
-
H1 (1.1)
fio^ra.
(a) Show that, to the lowest order in .1., the differeitial cross-section for two-
particle elastic scattering in the centre-of-mass frame is given by
do
72
de: (r.2)
128"4
where s - (pr * pz)' ,with p1 and p2being the momenta of the incoming particles.
(b) Use Wick's theorem to show that the graphs in Fig. 1.1 have the symmetry
factors as given. Also, check that these results agree with a compact expression
for the symmetry factor
s:B
n 2fl(n0"" (1.3)
n:2,3,...
where g is the number of possible permutations of vertices which leave unchanged
the diagram with fixed external lines, a, is the number of vertex pairs connected
by n identical lines, and p is the number of lines connecting a vertex with itself.
(c) Show that the two-point Green's function satisfies the relation
(n, + p\(0lr @@)Q0))10) : * QV(03(x)d0))10) - i6a(x- y).
Also verify this relation diagrammatically to first order in.l..
ry
s:4
Ftc. l.l. Symmetry factors.
