x
8.4 Scattering from a non-point-like object: the pion form factor
in e
− π
+
→ e
− π
+ . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.4.1 e
− scattering from a charge distribution . . . . . . . . 243
8.4.2 Lorentz invariance . . . . . . . . . . . . . . . . . . . . 244
8.4.3 Current conservation . . . . . . . . . . . . . . . . . . . 245
8.5 The form factor in the time-like region: e
+ e
−
→ π
+ π
− and
crossing symmetry . . . . . . . . . . . . . . . . . . . . . . . . 247
8.6 Electron Compton scattering . . . . . . . . . . . . . . . . . . 250
8.6.1 The lowest-order amplitudes . . . . . . . . . . . . . . 250
8.6.2 Gauge invariance . . . . . . . . . . . . . . . . . . . . . 251
8.6.3 The Compton cross section . . . . . . . . . . . . . . . 252
8.7 Electron muon elastic scattering . . . . . . . . . . . . . . . . 254
8.8 Electron–proton elastic scattering and nucleon form factors . 257
8.8.1 Lorentz invariance . . . . . . . . . . . . . . . . . . . . 258
8.8.2 Current conservation . . . . . . . . . . . . . . . . . . . 259
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9 Deep Inelastic Electron–Nucleon Scattering and the Parton
Model
269
9.1 Inelastic electron–proton scattering: kinematics and structure
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
9.2 Bjorken scaling and the parton model . . . . . . . . . . . . . 272
9.3 Partons as quarks and gluons . . . . . . . . . . . . . . . . . . 281
9.4 The Drell–Yan process . . . . . . . . . . . . . . . . . . . . . 284
9.5 e
+ e
− annihilation into hadrons . . . . . . . . . . . . . . . . . 288
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
IV Loops and Renormalization
297
10 Loops and Renormalization I: The ABC Theory
299
10.1 The propagator correction in ABC theory . . . . . . . . . . . 300
[2]
10.1.1 The O(g
2 ) self-energy Π (q
2
C
) . . . . . . . . . . . . . 300
10.1.2 Mass shift . . . . . . . . . . . . . . . . . . . . . . . . . 307
10.1.3 Field strength renormalization . . . . . . . . . . . . . 308
10.2 The vertex correction . . . . . . . . . . . . . . . . . . . . . . 311
10.3 Dealing with the bad news: a simple example . . . . . . . . . 314
[2]
10.3.1 Evaluating Π (q
2
C
) . . . . . . . . . . . . . . . . . . . . 314
10.3.2 Regularization and renormalization . . . . . . . . . . . 316
10.4 Bare and renormalized perturbation theory . . . . . . . . . . 318
10.4.1 Reorganizing perturbation theory . . . . . . . . . . . . 318
10.4.2 The O(g
2
ph ) renormalized self-energy revisited: how counter
terms are determined by renormalization conditions . 321
10.5 Renormalizability . . . . . . . . . . . . . . . . . . . . . . . . 324
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
8.4 Scattering from a non-point-like object: the pion form factor
in e
− π
+
→ e
− π
+ . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.4.1 e
− scattering from a charge distribution . . . . . . . . 243
8.4.2 Lorentz invariance . . . . . . . . . . . . . . . . . . . . 244
8.4.3 Current conservation . . . . . . . . . . . . . . . . . . . 245
8.5 The form factor in the time-like region: e
+ e
−
→ π
+ π
− and
crossing symmetry . . . . . . . . . . . . . . . . . . . . . . . . 247
8.6 Electron Compton scattering . . . . . . . . . . . . . . . . . . 250
8.6.1 The lowest-order amplitudes . . . . . . . . . . . . . . 250
8.6.2 Gauge invariance . . . . . . . . . . . . . . . . . . . . . 251
8.6.3 The Compton cross section . . . . . . . . . . . . . . . 252
8.7 Electron muon elastic scattering . . . . . . . . . . . . . . . . 254
8.8 Electron–proton elastic scattering and nucleon form factors . 257
8.8.1 Lorentz invariance . . . . . . . . . . . . . . . . . . . . 258
8.8.2 Current conservation . . . . . . . . . . . . . . . . . . . 259
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9 Deep Inelastic Electron–Nucleon Scattering and the Parton
Model
269
9.1 Inelastic electron–proton scattering: kinematics and structure
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
9.2 Bjorken scaling and the parton model . . . . . . . . . . . . . 272
9.3 Partons as quarks and gluons . . . . . . . . . . . . . . . . . . 281
9.4 The Drell–Yan process . . . . . . . . . . . . . . . . . . . . . 284
9.5 e
+ e
− annihilation into hadrons . . . . . . . . . . . . . . . . . 288
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
IV Loops and Renormalization
297
10 Loops and Renormalization I: The ABC Theory
299
10.1 The propagator correction in ABC theory . . . . . . . . . . . 300
[2]
10.1.1 The O(g
2 ) self-energy Π (q
2
C
) . . . . . . . . . . . . . 300
10.1.2 Mass shift . . . . . . . . . . . . . . . . . . . . . . . . . 307
10.1.3 Field strength renormalization . . . . . . . . . . . . . 308
10.2 The vertex correction . . . . . . . . . . . . . . . . . . . . . . 311
10.3 Dealing with the bad news: a simple example . . . . . . . . . 314
[2]
10.3.1 Evaluating Π (q
2
C
) . . . . . . . . . . . . . . . . . . . . 314
10.3.2 Regularization and renormalization . . . . . . . . . . . 316
10.4 Bare and renormalized perturbation theory . . . . . . . . . . 318
10.4.1 Reorganizing perturbation theory . . . . . . . . . . . . 318
10.4.2 The O(g
2
ph ) renormalized self-energy revisited: how counter
terms are determined by renormalization conditions . 321
10.5 Renormalizability . . . . . . . . . . . . . . . . . . . . . . . . 324
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326