Spin in Particle Physics

Annotation......Page 1
Series contents......Page 2
Title......Page 3
Date-line......Page 4
Contents ......Page 7
Preface ......Page 14
Acknowledgements ......Page 17
Notational conventions ......Page 18
1 Spin and helicity ......Page 22
1.1 Spin and rotations in non-relativistic quantum mechanics ......Page 23
1.2 Spin and helicity in a relativistic process ......Page 28
1.2.1 Particles with non-zero mass ......Page 32
1.2.2 The physical interpretation of helicity and canonical spin states ......Page 34
1.2.3 Particles with zero mass ......Page 36
2.1 Particles with non-zero mass ......Page 39
2.2.1 Pure rotation of axes ......Page 41
2.2.2 Pure Lorentz boost of axes ......Page 42
2.2.3 Boost along or opposite to $\mathbf{p}$ ......Page 43
2.2.5 Non-relativistic limit of CM to Lab transformation ......Page 44
2.2.8 The Thomas precession ......Page 45
2.3.1 Parity ......Page 48
2.3.2 Time reversal ......Page 49
2.4 Fields and wave functions ......Page 51
2.4.1 The discrete transformations of the fields ......Page 55
2.4.2 Isospin multiplets for antiparticles ......Page 57
3 The spin density matrix ......Page 59
3.1.1 Definition ......Page 60
3.1.2 Some general properties of $\rho_{mm'}$ ......Page 61
3.1.3 Combined systems of several particle types ......Page 63
3.1.4 The independent parameters specifying $\rho$ ......Page 64
3.1.5 The multipole parameters ......Page 65
3.1.6 Multipole parameters for combined systems of particles ......Page 67
3.1.8 The effect of rotations on the density matrix ......Page 68
3.1.9 Diagonalization of $\rho$. The quantization axis ......Page 69
3.1.10 Other choices of basis matrices ......Page 70
3.1.11 Invariant characterization of the state of polarization of an ensemble ......Page 71
3.1.12 Spin-1 particles and photons ......Page 72
3.1.13 Positivity of the density matrix ......Page 77
3.2.1 Definition of the helicity density matrix ......Page 78
3.2.2 Definition of helicity multipole parameters ......Page 79
3.2.4 Transformation law for multipole parameters ......Page 80
3.3 Choices of reference frame for a reaction ......Page 81
3.3.1 Density matrix for the initial particles ......Page 82
3.3.2 Density matrix of final state particles ......Page 84
3.4 Covariant spin vectors ......Page 88
4.1 Helicity amplitudes for elastic and pseudoelastic reactions ......Page 94
4.2.1 Parity ......Page 96
4.2.3 Identical particles ......Page 97
4.3 Some analytic properties of the helicity amplitudes ......Page 100
4.4 Crossing for helicity amplitudes ......Page 101
4.6.1 Matrix elements of a vector current ......Page 103
4.6.2 Vector and axial-vector coupling ......Page 106
4.6.3 Chirality ......Page 110
5.1 The generalized optical theorem ......Page 113
5.1.1 Nucleon-nucleon scattering ......Page 115
5.1.2 Particles of arbitrary spin ......Page 116
5.1.3 Application to deuteron-nucleon and deuteron-deuteron scattering ......Page 117
5.2.1 Definition ......Page 119
5.2.3 Angular momentum constraints near $\theta = 0, \pi$ ......Page 120
5.3 The CM observables and the dynamical reaction parameters ......Page 122
5.3.1 Properties of the CM reaction parameters ......Page 123
5.4 Experimental determination of the CM reaction parameters ......Page 125
5.4.1 Unpolarized initial state ......Page 126
5.4.2 Polarized beam, unpolarized target ......Page 128
5.4.4 Polarized beam and target ......Page 133
5.5 The laboratory reaction parameters ......Page 134
5.6.1 The reaction spin 1/2 + spin 1/2 -> spin 1/2 + spin 1/2 ......Page 136
5.6.3 The reactions spin 1/2 + spin 1/2 —> arbitrary-spin particles ......Page 141
5.7 Non-linear relations amongst the observables ......Page 144
5.8.1 CM reaction parameters and final state density matrix ......Page 146
6 The production of polarized hadrons ......Page 150
6.1 Polarized proton sources ......Page 151
6.2.1 Frozen targets ......Page 154
6.2.2 Gas-jet targets ......Page 158
6.3 The acceleration of polarized particles ......Page 164
6.3.1 Dynamics of the relativistic mean spin vector ......Page 165
6.3.2 Difficulties in the acceleration of polarized particles ......Page 168
6.3.3 The Siberian snake ......Page 172
6.3.4 Stern-Gerlach polarization of protons and antiprotons ......Page 175
6.4 Polarized secondary and tertiary beams ......Page 179
7.1 The natural polarization of electrons circulating in a perfect storage ring ......Page 186
7.1.1 Imperfect storage rings ......Page 194
7.2.1 Polarization at LEP ......Page 197
7.2.2 Polarization at HERA ......Page 200
7.3 Polarization at SLC ......Page 202
8 Analysis of polarized states: polarimetry ......Page 206
8.1 Stable particles ......Page 207
8.1.1 Reaction mechanism understood ......Page 208
8.1.2 Reaction mechanism not known ......Page 226
8.2.1 Two-particle decay of spin-$J$ resonance ......Page 230
8.2.2 Three-particle decay of a spin-$J$ resonance ......Page 244
9 Electroweak interactions ......Page 255
9.1 Summary of the Standard Model ......Page 256
9.2 Precision tests of the Standard Model ......Page 260
9.2.1 The reaction $e^-e^+$ —> fermion-antifermion pair ......Page 262
9.2.2 The reaction $e^-e^+$ —> quark-antiquark pair ......Page 276
9.3 Summary ......Page 278
10.1 A brief introduction to QCD ......Page 279
10.2 Local gauge invariance in QCD ......Page 283
10.3 Feynman rules for massless particles ......Page 285
10.3.1 The calculus of massless spinors ......Page 286
10.4 The helicity theorem for massless fermions ......Page 289
10.5 Spin structure from a fermion line ......Page 291
10.6 Example: high energy $e^- + \mu^-$ —> $e^- + \mu^-$ ......Page 293
10.7 Massive spinors ......Page 294
10.8 Polarization vectors ......Page 295
10.9 Shorthand notation for spinor products ......Page 299
10.10 QED: high energy Compton scattering ......Page 300
10.11 QCD: gluon Compton scattering ......Page 302
10.12 QCD: Multigluon amplitudes ......Page 305
10.12.1 The colour structure ......Page 306
10.12.2 Helicity structure of the n-gluon amplitude ......Page 307
10.12.3 The amplitude for $G + G$ —> $G + G$ ......Page 309
10.12.4 Colour sums for gluon reactions ......Page 313
10.12.5 Colour sum for $GG$ —> $GG$ ......Page 315
10.12.6 Some properties of $n$-gluon amplitudes ......Page 317
11.1 Introduction ......Page 319
11.2 Deep inelastic scattering ......Page 320
11.3 General formalism and structure functions ......Page 323
11.4 The simple parton model ......Page 326
11.5 Field-theoretic generalization of the parton model ......Page 329
11.5.1 Longitudinal polarization: the quark contribution to $g_1(x)$ ......Page 337
11.5.2 Transverse polarization: $g_2(x)$ ......Page 339
11.6 Moments of the structure functions, sum rules and the spin crisis ......Page 340
11.6.1 A spin crisis in the parton model ......Page 344
11.6.2 The gluon anomaly ......Page 345
11.7 QCD corrections and evolution ......Page 347
11.7.1 Beyond leading order; scheme dependence ......Page 348
11.8 Phenomenology: the polarized-parton distributions ......Page 352
11.8.1 Behaviour as $x$ —> 0 and $x$ —> 1 ......Page 357
11.9 The general partonic structure of the nucleon ......Page 360
11.9.1 Evolution for $\Delta_Tq(x, Q^2)$ ......Page 364
11.10 The future: neutrino beams? ......Page 365
12 Two-spin and parity-violating single-spin asymmetries at large scale ......Page 369
12.1 Inclusive and semi-inclusive reactions: general approach ......Page 370
12.2 Longitudinal two-spin asymmetries ......Page 373
12.2.1 $p p$ —> $\pi^0 X$ ......Page 376
12.2.2 Prompt photon production ......Page 378
12.2.3 The Drell-Yan reaction $p p$ —> $l^+ l^_- X$ ......Page 379
12.2.4 Drell-Yan production of $J/\Psi$ and $\chi_2$ ......Page 383
12.2.5 Semi-inclusive lepton-hadron scattering ......Page 386
12.3 Parity-violating longitudinal single-spin asymmetries ......Page 389
12.3.1 Small-$p_T$ single-spin $W^\pm$ and $Z^0$ production ......Page 390
12.3.2 Larger-$p_T$ single-spin $W^\pm$ production ......Page 392
12.4 Transverse two-spin asymmetries ......Page 396
13 One-particle inclusive transverse single-spin asymmetries ......Page 403
13.1 Theoretical approaches ......Page 405
13.2 Standard QCD-parton model with soft-physics asymmetries ......Page 409
13.3 Collins mechanism for single-spin asymmetry ......Page 412
13.4 Beyond the standard QCD parton model ......Page 419
13.5 Phenomenological models ......Page 429
13.5.1 The Lund model ......Page 430
13.5.2 The Thomas precession model ......Page 431
13.5.3 Concluding remarks ......Page 432
14 Elastic scattering at high energies ......Page 434
14.1 Small momentum transfer: general ......Page 435
14.2 Electromagnetic interference revisited ......Page 441
14.3 Elastic scattering at large momentum transfer ......Page 444
14.3.1 The asymptotic behaviour ......Page 445
14.3.2 Complications of exclusive reactions ......Page 448
14.3.3 Summary ......Page 452
Appendix 1 The irreducible representation matrices for the rotation group and the rotation functions $d^j_{\lambda\mu}(\theta)$ ......Page 454
A2.1 The finite-dimensional representations ......Page 458
A2.2 Spinors ......Page 460
A2.3 Connection between spinor and vector representations ......Page 463
A3.1 Relativistic quantum fields ......Page 465
A3.2 Irreducible relativistic quantum fields ......Page 467
A3.3 Parity and field equations ......Page 468
A3.4 The Dirac equation ......Page 470
A4.1 Definition of transversity amplitudes ......Page 471
A4.2 Symmetry of transversity amplitudes ......Page 472
A4.3 Some analytic properties of transversity amplitudes ......Page 473
Appendix 5 Common notations for helicity amplitudes ......Page 474
Appendix 6 The coefficients $\mathcal{A}_{l'm'}(lm)$ ......Page 476
Appendix 7 The coefficients $\mathcal{C}^{lm,l'm'}_{l_1m_1,l'_1m'_1}$ ......Page 478
A8.1 The CM reaction parameters ......Page 480
A8.2 The Argonne Lab reaction parameters ......Page 482
Appendix 9 'Shorthand' notation and nomenclature for the Argonne Lab reaction parameters ......Page 484
A10.1 0+1/2 —> 0+1/2 ......Page 486
A10.3 $A + B$ —> $A + B$ all with spin 1/2 ......Page 487
A10.4 Photoproduction of pseudoscalar mesons ......Page 490
A10.6 Baryon resonance production in $0^-(1/2)^+$ —> $0^-(3/2)^+$ ......Page 493
Appendix 11 The Feynman rules for QCD ......Page 494
A12.1 General properties ......Page 497
A12.2 Helicity spinors and Lorentz transformations ......Page 499
A12.3 The Dirac-Pauli representation ......Page 501
A12.5 Massless fermions ......Page 502
A12.6 The Fierz rearrangement theorem ......Page 504
References ......Page 506
Index ......Page 515
Dust cover......Page 522