The Standard Model in the Making: Precision Study of the Electroweak Interactions

Cover......Page 1
Title ......Page 2
Preface ......Page 4
Contents ......Page 8
1.1 QED Lagrangian, $U(1)$ invariance and Feynman rules ......Page 15
1.2 The processes $e^+ e^- \rightarrow \mu^+\mu^-, e^+e^-$ ......Page 16
1.2.1 The Born cross-sections ......Page 18
1.2.2 Polarized fermions ......Page 20
1.3 A list of QED one-loop diagrams ......Page 23
1.3.2 Fermionic self-energy ......Page 24
1.3.3 QED vertex ......Page 25
1.3.4 QED box diagrams ......Page 29
1.4.1 The basic approach to renormalization ......Page 31
1.4.2 On-shell versus \overline{MS} renormalization in QED ......Page 37
1.4.3 Parameter renormalization and the 5-matrix: the GMS framework ......Page 46
1.4.4 Renormalization and scale behaviour ......Page 49
1.4.5 Gauge dependence and renormalization ......Page 53
1.5 The dispersion relation for hadronic vacuum polarization ......Page 55
1.6 One-loop corrections and bremsstrahlung in $e^+ e^- \rightarrow f\bar f$ ......Page 56
1.6.1 Soft bremsstrahlung ......Page 57
1.6.2 The treatment of infrared divergences ......Page 58
1.6.3 Hard bremsstrahlung ......Page 63
1.6.4 Final state QED corrections with realistic cuts ......Page 65
1.7 Large logarithmic corrections in QED ......Page 69
1.7.1 Kinematics and cross-section for a $2 \rightarrow 2$ process ......Page 70
1.7.2 The structure function approach ......Page 71
1.7.3 The flux-function or radiator approach ......Page 75
1.8.1 The angular distribution ......Page 79
1.8.2 The integrated angular distribution ......Page 81
1.9 Initial-final interference corrections ......Page 82
2.2.1 General basis and notation ......Page 85
2.2.2 Initial state radiative corrections to narrow resonances ......Page 87
2.2.3 Folding with the energy resolution function ......Page 88
2.2.4 Determination of the width in the presence of radiative corrections ......Page 92
2.2.5 Parameters for a resonance with $\Gamma\gg\sigma_E$ ......Page 93
2.3.1 Soft photon approximation ......Page 95
2.3.2 Leading QED corrections around a resonance ......Page 99
2.3.3 The mixed and non-QED box diagrams ......Page 100
2.3.4 The box and soft photon corrections. Cancellation of infrared divergences ......Page 103
2.4 An inventory of QED corrections for a resonance ......Page 105
2.4.2 Bremsstrahlung corrections ......Page 106
2.4.3 YFS exponentiation ......Page 110
2.5 Bhabha scattering in the presence of a resonance ......Page 112
2.6 Helicity formalism ......Page 115
2.6.1 Polarization for spin = 1/2 particles ......Page 116
2.6.2 The example of $e^+ e^- \rightarrow \mu^+\mu^-\gamma$ ......Page 119
3.1.1 Lagrangian building ......Page 123
3.1.2 Tadpoles ......Page 133
3.2.1 The QCD Lagrangian ......Page 134
3.2.2 The \overline{MS} scheme and the running parameters ......Page 135
3.2.3 The four-loop improvement ......Page 140
3.3 Appendix: Feynman rules for vertices ......Page 141
4.1 Introduction: muon lifetime ......Page 147
4.2 Bremsstrahlung in muon decay ......Page 148
4.3 Virtual QED corrections ......Page 152
4.4 Total QED corrections ......Page 154
4.5 Two-loop QED corrections ......Page 155
5.1.1 One-point integrals, Л-functions ......Page 156
5.1.2 Two-point integrals, ?-functions ......Page 158
5.1.3 Derivatives of ^-functions ......Page 162
5.1.4 Three-point integrals, C-functions ......Page 164
5.1.5 Four-point integrals, J9-functions ......Page 173
5.1.6 a, b, c$^{(j)}$, d$^{(j)}$-functions ......Page 181
5.2 Bosonic self-energies and related topics ......Page 190
5.2.1 Vector-vector self-energies ......Page 192
5.2.2 Scalar-scalar self-energies ......Page 196
5.2.3 Scalar-vector transitions ......Page 202
5.2.4 A summary of results ......Page 204
5.2.5 Self-energy diagrams in the unitary gauge ......Page 211
5.2.6 A summary of results in the unitary gauge ......Page 214
5.3 Ward identities for self-energies and transitions ......Page 215
5.4 Appendix: total self-energies in the $R_\xi$ gauge ......Page 218
5.5.1 $\xi = 1$ gauge ......Page 223
5.5.2 $R_\xi$ gauge ......Page 224
5.5.3 Unitary gauge ......Page 227
5.6 Appendix: Bosonic self-energies and transitions ......Page 228
5.7 Fermion self-energy in the Standard Model ......Page 236
5.7.1 Fermionic self-energies in the $\xi = 1$ gauge ......Page 237
5.7.2 Fermionic self-energies in the $R_\xi$ gauge ......Page 238
5.7.4 Fermion wave-function renormalization ......Page 239
5.7.5 The $b$-quark wave-function renormalization ......Page 244
5.8 The Standard Model $Vf\bar f$ vertices ......Page 246
5.8.1 $Vf\bar f$ vertices in the 't Hooft-Feynman gauge and in the massless limit ......Page 247
5.8.2 The $m_t$-dependent part in $\xi = 1$ ......Page 253
5.8.3 The $Hb\bar b$ vertex in $\xi = 1$ ......Page 254
5.8.5 Infrared-divergent form factors ......Page 256
5.9 The $Vf\bar f$ vertices in the $R_\xi$ and unitary gauges ......Page 257
5.9.1 $\gamma f\bar f$-diagrams ......Page 259
5.9.2 $Z f\bar f$-diagrams ......Page 262
5.9.3 $W f\bar f$-diagrams ......Page 264
5.9.4 $H f\bar f$-diagrams ......Page 268
5.10.1 Taylor expansions for $Q^2 \rightarrow 0$ ......Page 270
5.10.2 $\gamma f\bar f$-diagrams ......Page 271
5.10.4 $W f\bar f$-diagrams ......Page 272
5.11.1 $(Z,A)$ vertices ......Page 273
5.11.2 $W$ vertices ......Page 274
5.11.3 $H$ vertices ......Page 275
5.12.1 General definitions ......Page 276
5.12.2 Calculation of a box diagram in the $\xi = 1$ gauge ......Page 278
5.12.3 A library of boxes for $NC$-processes ......Page 282
5.12.4 A library of boxes for $CC$ processes ......Page 287
5.13 Appendix: Box diagrams for $e^+e^-\rightarrow f\bar f$ ......Page 291
6.1 Introduction ......Page 292
6.3 The OMS schemes ......Page 294
6.4 The GMS scheme ......Page 300
6.5 Vector boson wave-function renormalization ......Page 301
6.6 Fermion wave-function renormalization ......Page 305
6.6.1 Equivalence to the OMS scheme ......Page 307
6.7 Renormalization with unstable particles ......Page 308
6.7.1 Fermion-loop approximation: massless fermions ......Page 309
6.7.3 Running couplings ......Page 310
6.7.4 Transition to the line-shape framework ......Page 315
6.8.1 Electric charge renormalization: the $\xi = 1$ gauge ......Page 317
6.8.2 Electric charge renormalization: the $R_\xi$ gauge ......Page 321
6.8.3 The OMS scheme in the unitary gauge ......Page 323
6.9 Muon decay ......Page 324
6.9.1 Muon decay: the $\xi = 1$ gauge ......Page 325
6.9.2 Muon decay: the $R_\xi$ gauge ......Page 329
6.10 Towards a choice of the input parameter set ......Page 330
6.11 One-loop in the GMS framework ......Page 332
6.11.1 Renormalization equations and their solution ......Page 334
6.11.2 The $Z$-propagator ......Page 336
6.11.3 Large effects in radiative corrections ......Page 337
6.11.4 Dyson re-summed propagators ......Page 340
6.11.5 Re-summation of large corrections: basics ......Page 343
6.11.6 Re-summation: improvements ......Page 346
6.12 The $W$ mass ......Page 354
6.12.1 What to use for $M_W$ in loops ......Page 356
6.12.2 $M_W$ and the LEP2 IPS ......Page 359
6.12.3 Vector boson propagators in the $t$-channel ......Page 360
6.13 The observable $\Delta r$ ......Page 362
6.13.1 The observable $\Delta r$ in the unitary gauge ......Page 364
6.13.3 Re-summation of large corrections ......Page 368
6.13.4 $\Delta r$ in the standard \overline{MS} scheme ......Page 370
6.14 One-loop renormalization and CKM mixing ......Page 373
6.15 Gauge parameter dependence: the building blocks ......Page 376
6.16 Tadpoles, $M_H$, renormalization and all that ......Page 382
6.16.1 $\Delta r$ and screening theorem ......Page 385
7.1 The partial widths of the $Z$ boson ......Page 387
7.1.1 The partial widths of $Z$ in the GMS approach ......Page 388
7.1.2 The partial widths of $Z$ in the OMS approach ......Page 395
7.2 The process $e^+e^-\rightarrow f\bar f$ ......Page 404
7.2.1 The Born amplitude and diagrams ......Page 405
7.2.2 The amplitude in the GMS approach ......Page 406
7.2.3 The amplitude in the OMS approach ......Page 412
8.2 Two-loop, heavy-top corrections ......Page 423
8.3 Mixed two-loop, QCD-weak corrections ......Page 425
8.3.1 Two-loop self-energies ......Page 426
8.3.2 Mixed QCD corrections in the OMS_scheme ......Page 428
8.3.3 Mixed two-loop effects in the $Z\rightarrow b\bar b$ vertex ......Page 429
8.3.4 More on QCD corrections to $\Delta\rho$ ......Page 430
8.4 Light quark production and QCD radiation ......Page 431
8.5 Sub-leading two-loop heavy-top corrections ......Page 432
8.5.1 Implementation of two-loop EW corrections within GMS approach ......Page 435
8.5.2 Implementation of two-loop EW corrections within the OMS approach ......Page 437
8.6.1 Parametric uncertainties ......Page 440
8.6.2 Theoretical uncertainties ......Page 441
8.7 Appendix: The two-loop, sub-leading ingredients ......Page 445
8.8 Appendix: Auxiliary functions for sub-leading corrections ......Page 448
9.1 The pseudo-observables ......Page 450
9.1.1 Pseudo-observables and experimental data ......Page 455
9.1.2 Fitting the experimental data ......Page 456
9.2 Realistic observables ......Page 459
9.3 The final state radiation of pairs in $e^+e^-$ annihilation ......Page 463
9.3.1 Hadron radiation ......Page 466
9.3.2 A summary of corrections to $Z$ decay into $l^+l^-$ ......Page 469
9.4 The initial state radiation of pairs ......Page 470
10.1 The partial widths of $W$ in the GMS approach ......Page 474
10.1.1 QED corrections: the massless approach ......Page 475
10.1.2 QCD corrections ......Page 483
10.2.1 One-loop corrections to the $W$ decay amplitude in the unitary gauge ......Page 484
10.2.2 The gauge invariance of the $W$ decay amplitude ......Page 486
10.2.3 The OMS $W$ width in one-loop approximation ......Page 487
11.1 Introduction ......Page 489
11.2.1 QED corrections ......Page 490
11.2.2 Full electroweak corrections in the GMS scheme ......Page 493
11.3.1 One-loop corrections to the $H$ decay amplitude ......Page 496
11.3.2 The OMS $H$ width in the one-loop approximation ......Page 498
11.3.3 The decay $H\rightarrow t\bar t$ ......Page 499
11.4 QCD corrections ......Page 500
11.5 The $H$ partial width into $ZZ$ ......Page 504
11.6 The $H$ partial widths into $\gamma\gamma(gg)$ ......Page 508
11.6.1 A summary of results for the MSM Higgs boson ......Page 510
12.1 QCD and the $Z$ decay ......Page 511
12.2 QCD radiator functions ......Page 515
12.3 Dimensional regularization and $\gamma_5$ ......Page 516
12.4 An inventory of QCD radiator functions ......Page 518
12.5 The cross-section $e^+e^-\rightarrow f\bar f$ and QCD ......Page 520
12.6 QCD corrections for the forward-backward asymmetry ......Page 521
13.1 Introduction ......Page 524
13.2 Born processes, $e^+e^-\rightarrow W^+W^-, ZZ, \gamma Z, \gamma\gamma, HZ$ ......Page 526
13.2.2 Lowest order helicity amplitudes ......Page 528
13.3 Narrow width approximation ......Page 531
14.2 Triple gauge boson couplings ......Page 533
14.2.1 Ward identities for trilinear couplings ......Page 535
14.2.2 Properties of the operators ......Page 537
14.2.3 The trilinear couplings made explicit ......Page 539
14.3.1 The wave-function of a vector boson ......Page 542
14.3.2 Treatment of internal fermion propagators ......Page 543
14.4 Complete and over-complete bases ......Page 545
14.5 Renormalization of the annihilation diagrams ......Page 549
14.5.1 Renormalization at $О(a)$ ......Page 550
14.5.2 The Complete Loop framework ......Page 551
14.5.3 Renormalization: back to the GMS scheme ......Page 555
14.6.1 Off-shell $\gamma(Z)e^+e^-$ and $We\nu$ vertices ......Page 560
14.7 A digression: $e^+e^-\rightarrow H\gamma$ ......Page 568
14.7.1 The off-shell $e^+e^-H$ vertex ......Page 571
14.8 Renormalization: $e^+e^-\rightarrow HZ$ ......Page 572
14.9 High-energy limit of vertex corrections ......Page 574
14.10 Box diagrams for $e^+e^-\rightarrow W^+W^-$ ......Page 577
14.11 Soft bremsstrahlung for $e^+e^-\rightarrow WW$ ......Page 579
14.12 Infrared limit of virtual corrections in $e^+e^-\rightarrow WW$ ......Page 584
14.12.1 The infrared limit of vertex diagrams ......Page 585
14.12.2 The infrared limit of the box diagrams ......Page 587
14.13 Appendix: complete collection of bosonic vertices ......Page 591
14.14 Appendix: Infrared limit of box diagrams ......Page 598
15.1 Introduction ......Page 602
15.2 Pair production away from the $Z$-resonance ......Page 603
15.3 Realistic distributions away from the $Z$-resonance ......Page 606
16.2 A classification of four-fermion processes ......Page 612
16.3.1 The fully extrapolated setup ......Page 616
16.3.2 The deterministic approach ......Page 623
16.4 Gauge invariance and the width of a vector boson ......Page 632
16.4.1 More Ward identities ......Page 638
16.5 The Fermion-Loop scheme and its applications ......Page 642
17.1 Introduction ......Page 648
17.2 Semi-leptonic channel $l\nu \rightarrow q\bar q$ ......Page 650
17.3 Fully-hadronic channel $q\bar q \rightarrow q\bar q$ ......Page 653
17.3.1 The CKM matrix ......Page 658
18.1 Introduction ......Page 659
18.2 The strategy of the calculation ......Page 662
18.3 Theoretical uncertainties ......Page 663
18.4 The Higgs signal and its background ......Page 665
18.5 Numerical results and comments ......Page 667
19 Appendix: some useful formulas ......Page 677
Bibliograpy ......Page 684
Index ......Page 693