viii
14.2.3 Local SU(3) c invariance: the QCD Lagrangian . . . .
82
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 ¯
pp collisions . . . . . . . . . . . . .
88
14.3.3 Three-jet events in ¯
pp 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 −q
2 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 inelastic scattering: scaling violations . . . . . . . . . . . . . . 135
15.6.1 Uncancelled mass singularities at order α s . . . . . . . 136
15.6.2 Factorization, and the order α s DGLAP equation . . 142
15.6.3 Comparison with experiment . . . . . . . . . . . . . . 145
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
16 Lattice Field Theory, and the Renormalization Group Revisited
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
14.2.3 Local SU(3) c invariance: the QCD Lagrangian . . . .
82
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 ¯
pp collisions . . . . . . . . . . . . .
88
14.3.3 Three-jet events in ¯
pp 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 −q
2 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 inelastic scattering: scaling violations . . . . . . . . . . . . . . 135
15.6.1 Uncancelled mass singularities at order α s . . . . . . . 136
15.6.2 Factorization, and the order α s DGLAP equation . . 142
15.6.3 Comparison with experiment . . . . . . . . . . . . . . 145
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
16 Lattice Field Theory, and the Renormalization Group Revisited
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