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Gauge Theory of Elementary Particle Physics
Article in Physics Today · December 1985
DOI: 10.1063/1.2814821
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Ta-Pei Cheng
Ling-Fong Li
University of Missouri - St. Louis
Chinese Academy of Sciences
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Gauge theory of
elementary particle
physics
Problems and solutions
TA-PEI CHENG
University of Missouri - St. Louis
and
LING-FONG LI
Carnegie Mellon University
CLARENDON PRESS
2000
•
OXFORD
Contents
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
Field quantization
_ Simple exercises in X(j>4 theory
Auxiliary
field
Disconnected diagrams
Simple external field problem
Path integral for a free particle
Path integral for a general quadratic action
Spreading of a wave packet
Path integral for a harmonic oscillator
Path integral for a partition function
Partition function for an SHO system
Non-standard path-integral representation
Weyl ordering of operators
Generating functional for a scalar
Poles in Green's function
1
field
6
8
9
11
13
16
17
21
23
25
26
32
35
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Renormalization
Counterterms in Xtp4 theory and in QED
Divergences in non-linear chiral theory
Divergences in lower-dimensional field theories
n-Dimensional 'spherical' coordinates
Some integrals in dimensional regularization
Vacuum polarization and subtraction schemes
Renormalization of X<p3 theory in n dimensions
Renormalization of composite operators
Cutkosky rules
37
39
41
43
46
49
53
57
59
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Renormalization group
Homogeneous renormalization-group equation
Renormalization constants
/3-function for QED
Behaviour of g near a simple fixed point
Running coupling near a general fixed point
One-loop renormalization-group equation in massless \<f>A theory
^-function for the Yukawa coupling
t
Solving the renormalization-group equation by Coleman's method
Anomalous dimensions for composite operators
63
64
67
69
70
71
72
75
77
viii
l
Contents
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
Group theory and the quark model
Unitary and hermitian matrices
SU(n) matrices
Reality of SU(2) representations
An identity for unitary matrices
An identity for SU(2) matrices
SU(3) algebra in terms of quark fields
Combining two spin-^ states
The SU(2) adjoint representation
Couplings of SU(2) vector representations
Isospin breaking effects
Spin wave function of three quarks
Permutation symmetry in the spin-isospin space
Combining two fundamental representations
SU(3) invariant octet baryon-meson couplings
Isospin wave functions of two pions
Isospins in non-leptonic weak processes
78
79
79
81
82
83
85
87
89
90
93
96
97
100
105
107
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
Chiral symmetry
Another derivation of Noether's current
Lagrangian with second derivatives
Conservation laws in a non-relativistic theory
Symmetries of the linear a-model
Spontaneous symmetry breaking in the a-model
PCAC in the a-model
Non-linear cr-model I
Non-linear a-model II
Non-linear a -model HI
SSB by two scalars in the vector representation
110
111
113
115
122
123
126
128
130
133
6
6.1
6.2
6.3
6.4
Renormalization and symmetry
Path-integral derivation of axial anomaly
Axial anomaly and r\ -*• yy
Soft symmetry breaking and renormalizability
Calculation of the one-loop effective potential
136
140
142
143
7
7.1
7.2
7.3
7.4
7.5
The Parton model and scaling
The Gottfried sum rule
Calculation of OPE Wilson coefficients
otot(e+e~ —*• hadrons) and short-distance physics
OPE of two charged weak currents
The total decay rate of the W-boson
/
146
147
151
155
156
8
8.1
8.2
8.3
Gauge symmetries
The gauge field in tensor notation
Gauge field and geometry
General relativity as a gauge theory
158
161
163
Contents
ix
8.4
8.5
8.6
8.7
8.8
8.9
O(n) gauge theory
Broken generators and Goldstone bosons
Symmetry breaking by an adjoint scalar
Symmetry breaking and the coset space
Scalar potential and first-order phase transition
Superconductivity as a Higgs phenomenon
165
167
169
171
172
173
9
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Quantum gauge theories
Propagator in the covariant R^ gauge
The propagator for a massive vector
Gauge boson propagator in the axial gauge
Gauge boson propagator in the Coulomb gauge
Gauge invariance of a scattering amplitude
Ward identities in QED
Nilpotent BRST charges
BRST charges and physical states
10
10.1
10.2
10.3
10.4
10.5
Quantum chromodynamics
Colour factors in QCD loops
Running gauge coupling in two-loop
Cross-section for three-jet events
Operator-product expansion of two currents
Calculating Wilson coefficients
188
191
193
198
201
11
11.1
11.2
11.3
11.4
11.5
11.6
11.7
11.8
Electroweak theory
Chiral spinors and helicity states
The polarization vector for a fermion
The pion decay rate and fn
Uniqueness of the standard model scalar potential
Electromagnetic and gauge couplings
Fermion mass-matrix diagonalization
An example of calculable mixing angles
Conservation of the B — L quantum number
205
206
208
212
213
214
215
216
12
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
Electroweak phenomenology
Atomic parity violation
Polarization asymmetry of Z - » / /
Simple r-lepton decays
Electron neutrino scatterings
CP properties of kaon non-leptonic decays
Z - • HH is forbidden
A/ — \ enhancement by short-distance QCD
Scalar interactions and the equivalence theorem
Two-body decays of a heavy Higgs boson
218
221
222
223
225
226
227
230
234
field
175
176
177
178
180
180
184
186
x
Contents
13
13.1
13.2
13.3
13.4
13.5
13.6
Topics in flavourdynamics
Anomaly-free condition in a technicolour theory
Pseudo-Goldstone bosons in a technicolour model
Properties of Majorana fermions
fj, —> ey and heavy neutrinos
Leptonic mixings in a vector-like theory
Muonium-antimuonium transition
238
239
239
244
250
252
14
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
Grand unification
Content of SU(5) representations
Higgs potential for SU(5) adjoint scalars
Massive gauge bosons in SU(5)
Baryon number non-conserving operators
SO(n) group algebra
Spinor representations of SO(«)
Relation between SO(2«) and SU(n) groups
Construction of SO(2n) spinors
255
256
258
260
260
263
267
269
15
15.1
15.2
15.3
15.4
Magnetic monopoles
The Sine-Gordon equation
Planar vortex
Stability of soliton
Monopole and angular momentum
275
280
282
283
16
16.1
16.2
16.3
Instantons
The saddle-point method
An application of the saddle-point method
A Euclidean double-well problem
field
289
292
295
References
301
Index
303
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