Pions and Nuclei (International Series of Monographs on Physics 74)

PIONS AND NUCLEI......Page 1
Title Page......Page 3
Copyright Page......Page 4
Preface......Page 6
Contents......Page 8
1.1 Constituents of the atomic nucleus......Page 18
1.2.1 The pion form factor......Page 19
1.2.2 Role of the ρ meson......Page 21
1.2.3 Why is the pion special?......Page 22
1.3 Size of the proton......Page 23
References......Page 24
2.1 Symmetries of strong interactions......Page 25
2.2 The free-pion field......Page 26
2.3.1 Static pion field from a point nucleon source......Page 28
2.3.2 Pseudoscalar and pseudovector coupling......Page 29
2.4.1 Variables and amplitudes......Page 30
2.4.2 Empirical properties of pion–nucleon scattering......Page 33
2.5 Phenomenological model of p-wave πN-scattering......Page 38
2.5.1. The p-wave Born terms for static nucleons......Page 39
2.5.2. The Δ(1232)-isobar model......Page 42
2.5.3 Relativistically improved isobar model......Page 45
2.5.4 Crossed Born terms and the \'range\' of the πN interaction......Page 49
2.6.1 The phenomenological s-wave Hamiltonian......Page 50
2.6.2 The s-wave threshold expansion......Page 52
2.6.3 The ρ-meson exchange model......Page 53
2.7 Summary......Page 54
Notes and further reading......Page 55
References......Page 56
3.1.1 The Yukawa picture......Page 57
3.1.3 Characteristic domains of the nucleon–nucleon interactions......Page 58
3.2.1 The OPE potential......Page 59
3.2.3 Comparison with magnetic dipole–dipole interactions......Page 60
3.3.2 Strength of the central OPE potential......Page 61
3.3.4 Momentum space representation of the static OPE potential......Page 62
3.3.6 Generalization of OPE to the NΔ-system......Page 63
3.4.2 The deuteron s- and d-wave functions......Page 65
3.4.4 Quadrupole moment......Page 67
3.5 The deuteron coupled equations and OPE......Page 68
3.6 Deuteron properties and OPE......Page 69
3.7.1 Formal solution and Green functions of the coupled equations......Page 71
3.7.2 The asymptotic d/s ratio......Page 72
3.7.3 The quadrupole moment......Page 73
3.8 The pionic Born terms......Page 75
3.8.1 P-wave scattering volumes......Page 76
3.8.2 Role of OPE in high nucleon–nucleon partial waves......Page 78
3.9.1 Boson–nucleon effective Lagrangians......Page 80
3.9.2 OBE potentials......Page 81
3.9.3 Structure of the scalar and vector exchange potentials......Page 82
3.9.4 Physical interpretation of effective bosons......Page 83
3.9.5 Discussion and summary of OBE models......Page 84
3.10.1 Introduction......Page 86
3.10.2 Construction of the two-pion exchange potential......Page 87
3.10.3 The ππ mass distribution......Page 89
3.10.4 Scalar–isoscalar exchange......Page 90
3.10.5 Vector–isovector exchange......Page 92
3.10.6 Role of the Δ(1232) in the two-pion exchange interaction......Page 93
3.10.7 The Paris potential: an example of the dispersive approach......Page 94
3.10.8 The isovector tensor potential......Page 96
3.11.1 The forward dispersion relation for potential scattering......Page 97
3.11.2 Potential scattering of identical particles......Page 98
3.11.4 Forward dispersion relations with OPE......Page 99
3.11.5 The discrepancy function Δ(z)......Page 100
3.11.6 An example: determination of the πºNN coupling constant......Page 103
3.11.7 Experimental evidence for exchanges in the NN force......Page 105
3.11.8 Summary of discrepancy function results......Page 109
3.12 The parity non-conserving NN potential......Page 110
3.12.2 The parity non-conserving OPE......Page 111
3.13 Summary......Page 113
Notes and further reading......Page 114
References......Page 115
4.2.1 The impulse approximation......Page 117
4.2.2 The πd forward scattering amplitude......Page 119
4.2.4 The double-scattering correction......Page 120
4.2.5 The real part of the πd forward amplitude......Page 122
4.2.6 Constraints on the deuteron from the πd cross-section......Page 123
4.3 Charge symmetry violation in πd scattering......Page 124
4.4 The pion–deuteron scattering length......Page 127
4.5.1 General considerations......Page 129
4.5.2 The pion–deuteron system......Page 130
4.5.3 πd ↔ NN data and phenomenology......Page 132
4.6.1 The impulse approximation......Page 136
4.6.2 The s-wave rescattering mechanism......Page 138
4.6.3 The p-wave rescattering mechanism......Page 140
4.6.4 Limitations of the rescattering model......Page 143
4.7.2 Qualitative dynamical features......Page 144
4.7.3 Inelasticities in nucleon–nucleon scattering......Page 145
4.8 Three-body approach to the πNN system......Page 147
4.8.1 Outline of three-body theory......Page 148
4.8.2 Applications to the pion–deuteron system......Page 150
4.8.3 Advantages and limitations of the three-body approach......Page 151
References......Page 152
5.1 Optical analogues......Page 154
5.2.1 Scattering from a single dipole......Page 155
5.2.2 Scattering from a system of dipoles......Page 156
5.2.3 The Lorentz–Lorenz correction: elementary derivation......Page 157
5.2.4 The Lorentz–Lorenz effect: pair-correlation approach......Page 159
5.3.1 The basic scattering equations......Page 161
5.3.2 The s-wave effective field......Page 162
5.3.3 Systems with two types of s-wave scatterers......Page 163
5.4.1 The p-wave π–nuclear potential......Page 164
5.4.2 The s-wave π-nuclear potential......Page 165
5.5 Pion optics: index of refraction and mean free path......Page 166
5.6 The spectral branches of nuclear pion physics......Page 169
5.7.1 Basic Fermi gas properties......Page 170
5.7.2 The pion self-energy......Page 171
5.7.3 Lowest-order p-wave pion self-energy: nucleon terms......Page 172
5.7.4 Lowest-order p-wave pion self-energy: Δ(1232) terms......Page 176
5.8 Spin–isospin sound in neutron matter: a schematic model......Page 178
5.9.1 Basic types of spin–isospin excitations......Page 181
5.9.2 Structure of the nuclear spin–isospin interaction......Page 182
5.9.3 Phenomenology of short-range spin–isospin correlations......Page 183
5.9.4 Relation to nuclear Fermi liquid theory......Page 185
5.9.5 Generalized Lorentz–Lorenz correction......Page 188
5.10 The diamesic function and the pionic response......Page 189
5.11 The static diamesic function and precursor effects......Page 191
5.12.1 Retrospective......Page 192
5.12.2 Conditions for pion condensation......Page 193
5.13 Pion-like excitations in nuclear matter: a résumé......Page 198
Notes and further reading......Page 199
References......Page 200
6.1 Formation and qualitative features......Page 202
6.2.1 Basic equation......Page 204
6.2.2 Bound state energies......Page 205
6.2.5 Tests of the pionic Klein–Gordon equation......Page 206
6.2.6 Behaviour for large Z......Page 208
6.3.2 Strong interaction energy shifts and scattering lengths......Page 209
6.3.3 Empirical properties of shifts and widths......Page 211
6.3.4 Estimates of level shifts......Page 213
6.3.5 Effect of finite nuclear size......Page 215
6.3.6 S-wave shifts and scattering lengths in very light nuclei......Page 217
6.4.1 The optical potential to leading order......Page 219
6.4.3 Kinematic corrections......Page 220
6.4.4 The complete optical potential at threshold......Page 221
6.5.1 Empirical values......Page 222
6.5.2 Significance of the phenomenological parameters......Page 224
6.6.1 Pion states in an infinite spherical potential well......Page 226
6.6.2 Bound state solutions: physical constraints and properties......Page 228
Notes and further reading......Page 229
References......Page 230
7.1 Introduction......Page 232
7.2.1 The Born approximation......Page 233
7.2.2 A kinematical effect: the angle transformation......Page 235
7.2.3 The optical potential at low energy......Page 237
7.2.4 Reaction cross-sections and the optical potential......Page 240
7.3 Phenomenology of elastic scattering in the Δ(1232) region......Page 243
7.3.1 Total cross-sections......Page 244
7.3.2 Angular distributions for elastic scattering......Page 245
7.3.3 Pion–nucleus partial wave amplitudes......Page 247
7.3.4 Pion–nucleus forward dispersion relations......Page 250
7.4.1 Manifestations of the Δ(1232) in nuclei......Page 254
7.4.2 The Δ-hole model......Page 255
7.4.3 Coherent multiple scattering in the Δ-hole model......Page 259
7.4.4 Δ-hole doorway states......Page 262
7.4.5 Summary: the Δ(1232) as a quastparticle......Page 266
7.5.1 Scattering to discrete nuclear states......Page 268
7.5.2 Quasifree scattering......Page 271
7.6.1 General features......Page 273
7.6.2 Single-charge exchange reactions......Page 276
7.6.3 Double-charge exchange reactions......Page 278
7.7.1 Introduction and kinematical considerations......Page 282
7.7.3 Implications of two-nucleon absorption models......Page 283
7.7.4 An example: π-3He absorption......Page 285
7.8 Summary......Page 287
Notes and further reading......Page 288
References......Page 289
8.1 Introduction......Page 291
8.2.1 Qualitative features of cross-sections......Page 292
8.2.2 Electric and magnetic dipole amplitudes......Page 294
8.2.3 Low-energy limit of pion photoproduction......Page 297
8.2.4 Currents and effective Hamiltonian of the γπN system......Page 298
8.2.5 Born terms for pion photoproduction......Page 301
8.2.6 The Δ(1232) excitation......Page 304
8.3.1 The exchange potential and its two-body currents......Page 306
8.3.2 The one-pion exchange current......Page 308
8.3.3 The Δ(1232) exchange current......Page 310
8.4 Siegert\'s theorem......Page 311
8.5 Magnetic exchange current phenomena......Page 312
8.5.1 The exchange magnetic moment......Page 313
8.5.2 Meson exchange current effects in np → dγ......Page 314
8.5.3 Exchange currents in deuteron electrodisintegration......Page 318
8.5.4 Exchange currents and the magnetic form factors of 3He and 3H......Page 320
8.5.5 Renormalization of the orbital g-factor......Page 322
8.6.1 The dipole sum rule......Page 325
8.6.2 The deuteron dipole sum rule......Page 327
8.6.3 Dipole sum rule enhancement in complex nuclei......Page 329
8.6.4 Relation between the dipole sum rule and δg l......Page 332
8.7 Nuclear photopion reactions near threshold......Page 333
8.7.1 Charged pion processes......Page 334
8.7.2 Neutral pion production......Page 335
8.8.1 General features......Page 337
8.8.2 Photon–nucleus scattering the Δ-hole model......Page 338
8.8.3 The (γ, πº) reaction in the Δ(1232) region......Page 342
Notes and further reading......Page 343
References......Page 346
9.1.1 An example: soft photons and Thomson scattering......Page 348
9.2.1 Currents and weak interaction phenomenology......Page 349
9.2.2 The charged pion decay......Page 350
9.3.2 A QCD sketch......Page 351
9.3.3 QCD with massless quarks: chiral symmetry......Page 352
9.3.4 A two-phase picture......Page 354
9.3.5 The nucleon axial current: a first approach......Page 355
9.3.7 PCAC and the Goldberger–Treiman relation......Page 356
9.4.1 Nucleon axial current: general structure......Page 357
9.4.2 The induced pseudoscalar current......Page 358
9.4.3 S-wave photoproduction of soft pions......Page 360
9.4.4 Pion–nucleon scattering lengths......Page 361
9.4.5 Soft-pion production by the axial current......Page 364
9.4.6 Axial p-wave pion production......Page 365
9.5.1 Pion–nuclear scattering lengths......Page 367
9.5.2 Nuclear pion photoproduction at threshold......Page 369
9.5.3 S-wave π absorption and production......Page 370
9.6.1 Introduction......Page 371
9.6.2 PCAC and the nuclear axial current......Page 372
9.6.3 The axial pion-exchange current......Page 373
9.6.4 Physical intepretation of the axial current......Page 375
9.7.1 The time component: O^+ ↔ O^– transitions in the A = 16 system......Page 376
9.7.2 The space component: the A = 3 system......Page 378
9.7.3 The pp → de^+ ν reaction......Page 379
9.7.4 Deuteron electrodisintegration revisited......Page 380
9.8 High-energy neutrino reactions......Page 382
Notes and further reading......Page 384
References......Page 386
10.1 Introduction......Page 388
10.2 Basic spin–isospin operators and transitions......Page 389
10.3.1 Gross features of the nucleon–nucleon T-matrix......Page 390
10.3.2 The np charge exchange reaction......Page 393
10.4.1 The (p, n) reaction in nuclei......Page 394
10.4.2 Systematics of Gamow–Teller strength distributions......Page 395
10.4.3 Schematic model of the Gamow–Teller resonance......Page 397
10.5 Δ-isobar excitations......Page 400
10.5.1 The p(3He, 3H)Δ^++ reaction......Page 401
10.6. The nuclear spin–isospin response function......Page 402
10.6.1 Linear response: general framework......Page 403
10.6.2 Schematic picture of nuclear spin–isospin response......Page 404
10.6.3 The RPA framework......Page 406
10.7 An example: pion-like 2^– states in 16O......Page 409
10.8 Renormalization of spin–isospin operators......Page 410
10.8.1 Mechanisms for quenching g A......Page 411
10.8.2 Renormalization of the spin g-factor......Page 413
10.8.3 An example: M1 transition in 48Ca......Page 414
10.9 Conclusions......Page 415
Notes and further reading......Page 416
References......Page 418
Appendix 1. Four-vectors: notations and metric conventions......Page 420
(b) Dirac matrices......Page 421
(b) Pions......Page 423
(c) Δ-isobars......Page 424
(e) Isospin projection operators......Page 425
(g) G-parity......Page 426
(a) Free-pion field......Page 427
(b) Free Dirac fields......Page 429
(c) The free spin 3/2 field......Page 430
References......Page 432
(a) Klein–Gordon propagators......Page 433
(b) Dirac propagators......Page 435
References......Page 436
(d) Divergence of the Dirac axial current......Page 437
(a) Nucleon electromagnetic form factors......Page 440
(b) Nucleon axial form factors......Page 442
(c) Pion electromagnetic form factor......Page 443
References......Page 444
(b) Invariant amplitudes for πN scattering......Page 446
(d) Relation between the T-matrix and the differential cross-section......Page 447
(f) Partial wave expansion......Page 448
(g) Projection operator relations......Page 449
Reference......Page 450
(c) Amplitudes......Page 451
(d) Differential cross-section......Page 452
References......Page 453
(b) Definition of potential V......Page 454
(d) Pseudoscalar, scalar, and vector exchange potentials......Page 455
(e) Matrix elements of the tensor operator S 12......Page 457
References......Page 458
(b) Isospin crossing relations......Page 459
(c) Fierz transformation......Page 460
(a) Basic assumptions......Page 461
(c) Subtractions......Page 462
(d) Crossing relations......Page 463
(f) The unphysical region (|ω| < m)......Page 464
(g) Non-relativistic potential scattering......Page 465
References......Page 466
(a) Interaction Hamiltonian......Page 467
(c) Non-relativistic reduction......Page 468
References......Page 469
(a) General features......Page 470
(b) Currents and their algebra......Page 471
(d) The non-linear σ model......Page 472
References......Page 474
(b) Lindhard function ϕ(ω, k)......Page 475
Reference......Page 477
(d) Limit for small z......Page 478
(h) Recursion relations......Page 479
(k) Outgoing wave expansion......Page 480
(b) Legendre functions of second kind......Page 481
Author Index......Page 484
Subject Index......Page 490