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    Gauge theory ofelementary particlephysicsProblems and solutionsTA-PEI CHENGUniversity ofMissouri - St. LouisandL I N G - F O N G LICarnegie Mellon University

    CLARENDON PRESS OXFORD2000

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    Contents

    1 Field quantization1.1 _ Simple exercises in X(j> 4 theory 11.2 Auxiliary field 61.3 Disconnected diagrams 81.4 Simple external field problem 91.5 Path integral for a free partic le 111.6 Path integral for a general quadratic action 131.7 Spreading of a wave packet 161.8 Path integral for a harm onic osc illator 171.9 Path integral for a partition function 211.10 Partition function for an SHO system 231.11 Non -standard path-integral representation 251.12 Weyl ordering of opera tors 261.13 Generating functional for a scalar field 321.14 Poles in G ree n's function 352 Renormalization2.1 Counterterms in Xtp 4 theory and in QED 372.2 Divergences in non-linear chiral theory 392.3 Divergences in lower-dimensional field theories 412.4 n-Dimensional 'sphe rical' coordinates 432.5 Some integrals in dimension al regularization 462.6 Vacuum polarization and subtraction schem es 492.7 Renormalization of X

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    viii l Contents4 Group theory and the quark model4.1 Unitary and hermitian matrices 784.2 SU(n) matrices 794.3 Reality of SU(2) representations 794.4 An identity for unitary matrices 814.5 An identity for SU(2) ma trices 824.6 SU(3) algebra in terms of quark fields 834.7 Com bining two spin-^ states 854.8 The SU(2) adjoint representation 874.9 Coup lings of SU(2) vector representations 894.10 Isospin breaking effects 904.11 Spin wave function of three quarks 934.12 Permutation symmetry in the spin-isos pin space 964.13 Com bining two fundamental representations 974.14 SU(3) invariant octet bary on-m eson couplings 1004.15 Isospin wave functions of two pions 1054.16 Isospins in non-leptonic weak processes 1075 Chiral symmetry5.1 Ano ther derivation of Noethe r's current 1105.2 Lagrangian with second derivatives 1115.3 Conservation laws in a non-relativistic theory 1135.4 Sym metries of the linear a-m od el 1155.5 Spontaneous symmetry breaking in the a-m od el 1225.6 PCAC in the a-m od el 1235.7 Non-linear cr-model I 1265.8 Non-linear a-m od el II 1285.9 Non-linear a -model HI 1305.10 SSB by two scalars in the vector representation 1336 Renormalization and symmetry6.1 Path-integral derivation of axial anom aly 1366.2 Axial anom aly and r\ -* yy 1406.3 Soft symmetry breaking and renorm alizability 1426.4 Calculation of the one-loop effective potential 1437 The Parton model and scaling7.1 The Gottfried sum rule 1467.2 Calcu lation of OPE W ilson coefficients 1477.3 o to t(e+e~ * hadrons) and short-distance physics 1517.4 OPE of two charged weak currents 1557.5 The total decay rate of the W-boson / 1568 Gauge symmetries8.1 The gauge field in tensor notation 1588.2 Gauge field and geom etry 1618.3 General relativity as a gauge theory 163

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    Contents ix8.4 O(n) gaug e theory 1658.5 Broken genera tors and Goldstone bosons 1678.6 Symmetry breaking by an adjoint scalar 1698.7 Sym metry breaking and the coset space 1718.8 Scalar poten tial and first-order phase transition 1728.9 Superconductivity as a Higgs phenomenon 1739 Quantum gauge theories9.1 Propagator in the covariant R^ gauge 1759.2 The propagator for a massive vector field 1769.3 Gauge boson propaga tor in the axial gaug e 1779.4 Gauge boson propagator in the Coulom b gauge 1789.5 Gauge invariance of a scattering amplitude 1809.6 Ward identities in QED 1809.7 Nilpotent BR ST charges 1849.8 BRST charges and physica l states 18610 Quantum chromodynamics10.1 Colour factors in QC D loops 18810.2 Running gauge coup ling in two-loo p 19110.3 Cross-section for three-jet events 19310.4 Operator-product expansion of two currents 19810.5 Calculating Wilson coefficients 20111 Electroweak theory11.1 Chiral spinors and helicity states 20511.2 The polarization vector for a fermion 20611.3 The pion decay rate and fn 20811.4 Uniqueness of the standard model scalar potential 21211.5 Electromagnetic and gau ge coup lings 21311.6 Fermion mass-matrix diagonalization 21411.7 An exam ple of calcu lable mixing angles 21511.8 Conservation of the B L quantum number 21612 Electroweak phenomenology12.1 Atom ic parity violation 21812.2 Polarization asymmetry of Z - / / 22112.3 Simple r-lepto n decay s 22212.4 Electron neutrino scatterings 22312.5 CP properties of kaon non-lepton ic decays 22512.6 Z - HH is forbidden 22612.7 A / \ enhancement by short-distance QC D 22712.8 Scalar interactions and the equivalence theorem 23012.9 Two-body decays of a heavy Higgs boson 234

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    x Contents13 Topics in flavourdynamics13.1 Anomaly-free condition in a technicolour theory 23813.2 Pseudo-G oldstone bosons in a technicolour model 23913.3 Prope rties of Majorana fermions 23913.4 fj ,> ey and heavy neutrinos 24413.5 Leptonic mixings in a vector-like theory 25013.6 M uonium -antimu onium transition 25214 Grand unification14.1 Con tent of SU (5) represen tations 25514.2 Higgs potential for SU (5) adjoint scalars 25614.3 M assive gauge bosons in SU(5) 25814.4 Baryon number non-conserving operators 26014.5 SO (n) group algebra 26014.6 Spinor representations of SO () 26314.7 Relation between SO (2) and SU (n) groups 26714.8 Construction of SO (2n) spinors 26915 Magnetic monopoles15.1 The Sine -G ordo n equation 27515.2 Planar vortex field 28015.3 Stability of soliton 28215.4 M onopole and angular mom entum 28316 Instantons16.1 The saddle-po int method 28916.2 An application of the saddle-point method 29216.3 A Euclidean double-well problem 295References 301Index 303