Gauge Theory of Elementary Particle Physics Problems and Solutions

by Cheng, Ta-Pei; Li, Ling-Fong

• ISBN13: 9780198506218
• ISBN10: 019850621X
• Format: Paperback
• Copyright: 2000-05-04
• Publisher: Oxford University Press

Summary

Students of particle physics often find it difficult to locate resources to learn calculational techniques as intermediate steps are usually not given in the research literature. To a certain extent, this is the case even in some of the established textbooks. In this book of worked problems enough details are provided so that the beginner will understand the solution in each particular case. With this step-by-step guidance, students (after first attempting the solution themselves) can develop their skills and confidence in their ability to work out particle theory problems. Besides being a problems/solutions addition to the well established textbook by Cheng & Li, this book introduces several new topics. It provides the reader with a self-contained approach to the subject, suitable even for those not familiar with the textbook. All problems have been given a descriptive title, enabling the reader to select according to his preferences.

Table of Contents

Field quantization
Simple exercises in λø4 theory	1	(5)
Auxiliary field	6	(2)
Disconnected diagrams	8	(1)
Simple external field problem	9	(2)
Path integral for a free particle	11	(2)
Path integral for a general quadratic action	13	(3)
Spreading of a wave packet	16	(1)
Path integral for a harmonic oscillator	17	(4)
Path integral for a partition function	21	(2)
Partition function for an SHO system	23	(2)
Non-standard path-integral representation	25	(1)
Weyl ordering of operators	26	(6)
Generating functional for a scalar field	32	(3)
Poles in Green's function	35	(2)
Renormalization
Counterterms in λø4 theory and in QED	37	(2)
Divergences in non-linear chiral theory	39	(2)
Divergences in lower-dimensional field theories	41	(2)
n-Dimensional `spherical' coordinates	43	(3)
Some integrals in dimensional regularization	46	(3)
Vacuum polarization and subtraction schemes	49	(4)
Renormalization of λø3 theory in n dimensions	53	(4)
Renormalization of composite operators	57	(2)
Cutkosky rules	59	(4)
Renormalization group
Homogeneous renormalization-group equation	63	(1)
Renormalization constants	64	(3)
β-function for QED	67	(2)
Behaviour of g near a simple fixed point	69	(1)
Running coupling near a general fixed point	70	(1)
One-loop renormalization-group equation in massless λø4 theory	71	(1)
β-function for the Yukawa coupling	72	(3)
Solving the renormalization-group equation by Coleman's method	75	(2)
Anomalous dimensions for composite operators	77	(1)
Group theory and the quark model
Unitary and hermitian matrices	78	(1)
SU(n) matrices	79	(1)
Reality of SU(2) representations	79	(2)
An identity for unitary matrices	81	(1)
An identity for SU(2) matrices	82	(1)
SU(3) algebra in terms of quark fields	83	(2)
Combining two spin 1/2 states	85	(2)
The SU(2) adjoint representation	87	(2)
Couplings of SU(2) vector representations	89	(1)
Isospin breaking effects	90	(3)
Spin wave function of three quarks	93	(3)
Permutation symmetry in the spin-isospin space	96	(1)
Combining two fundamental representations	97	(3)
SU(3) invariant octet baryon-meson couplings	100	(5)
Isospin wave functions of two pions	105	(2)
Isospins in non-leptonic weak processes	107	(3)
Chiral symmetry
Another derivation of Noether's current	110	(1)
Lagrangian with second derivatives	111	(2)
Conservation laws in a non-relativistic theory	113	(2)
Symmetries of the linear σ-model	115	(7)
Spontaneous symmetry breaking in the σ-model	122	(1)
PCAC in the σ-model	123	(3)
Non-linear σ-model I	126	(2)
Non-linear σ-model II	128	(2)
Non-linear σ-model III	130	(3)
SSB by two scalars in the vector representation	133	(3)
Renormalization and symmetry
Path-integral derivation of axial anomaly	136	(4)
Axial anomaly and η &arrow; γγ	140	(2)
Soft symmetry breaking and renormalizability	142	(1)
Calculation of the one-loop effective potential	143	(3)
The Parton model and scaling
The Gottfried sum rule	146	(1)
Calculation of OPE Wilson coefficients	147	(4)
σtot(e+e- &arrow; hadrons) and short-distance physics	151	(4)
OPE of two charged weak currents	155	(1)
The total decay rate of the W-boson	156	(2)
Gauge symmetries
The gauge field in tensor notation	158	(3)
Gauge field and geometry	161	(2)
General relativity as a gauge theory	163	(2)
O(n) gauge theory	165	(2)
Broken generators and Goldstone bosons	167	(2)
Symmetry breaking by an adjoint scalar	169	(2)
Symmetry breaking and the coset space	171	(1)
Scalar potential and first-order phase transition	172	(1)
Superconductivity as a Higgs phenomenon	173	(2)
Quantum gauge theories
Propagator in the covariant Rξ gauge	175	(1)
The propagator for a massive vector field	176	(1)
Gauge boson propagator in the axial gauge	177	(1)
Gauge boson propagator in the Coulomb gauge	178	(2)
Gauge invariance of a scattering amplitude	180	(1)
Ward identities in QED	180	(4)
Nilpotent BRST charges	184	(2)
BRST charges and physical states	186	(2)
Quantum chromodynamics
Colour factors in QCD loops	188	(3)
Running gauge coupling in two-loop	191	(2)
Cross-section for three-jet events	193	(5)
Operator-product expansion of two currents	198	(3)
Calculating Wilson coefficients	201	(4)
Electroweak theory
Chiral spinors and helicity states	205	(1)
The polarization vector for a fermion	206	(2)
The pion decay rate and fπn	208	(4)
Uniqueness of the standard model scalar potential	212	(1)
Electromagnetic and gauge couplings	213	(1)
Fermion mass-matrix diagonalization	214	(1)
An example of calculable mixing angles	215	(1)
Conservation of the B-L quantum number	216	(2)
Electroweak phenomenology
Atomic parity violation	218	(3)
Polarization asymmetry of Z &arrow; ff	221	(1)
Simple τ-lepton decays	222	(1)
Electron neutrino scatterings	223	(2)
CP properties of kaon non-leptonic decays	225	(1)
Z &arrow; HH is forbidden	226	(1)
ΔI = 1/2 enhancement by short-distance QCD	227	(3)
Scalar interactions and the equivalence theorem	230	(4)
Two-body decays of a heavy Higgs boson	234	(4)
Topics in flavourdynamics
Anomaly-free condition in a technicolour theory	238	(1)
Pseudo-Goldstone bosons in a technicolour model	239	(1)
Properties of Majorana fermions	239	(5)
μ ey and heavy neutrinos	244	(6)
Leptonic mixings in a vector-like theory	250	(2)
Muonium-antimuonium transition	252	(3)
Grand unification
Content of SU(5) representations	255	(1)
Higgs potential for SU(5) adjoint scalars	256	(2)
Massive gauge bosons in SU(5)	258	(2)
Baryon number non-conserving operators	260	(1)
SO(n) group algebra	260	(3)
Spinor representations of SO(n)	263	(4)
Relation between SO(2n) and SU(n) groups	267	(2)
Construction of SO(2n) spinors	269	(6)
Magnetic monopoles
The Sine-Gordon equation	275	(5)
Planar vortex field	280	(2)
Stability of soliton	282	(1)
Monopole and angular momentum	283	(6)
Instantons
The saddle-point method	289	(3)
An application of the saddle-point method	292	(3)
A Euclidean double-well problem	295	(6)
References	301	(2)
Index	303