Table of Contents
Preface v
1 Introduction 1
1.1 Non-relativistic three-nucleon and three-quark systems 1
1.1.1 Description of three-nucleon systems 2
1.1.2 Three-quark systems 3
1.2 Dispersion relation technique for three particle systems 4
1.2.1 Elements of the dispersion relation technique for two-particle systems 5
1.2.2 Interconnection of three particle decay amplitudes and two-particle scattering ones in hadron physics 6
1.2.3 Quark-gluon language for processes in regions I, III and IV 10
1.2.4 Spectral integral equation for three particles 11
1.2.5 Isobar models 12
1.2.6 Quark-diquark model for baryons and group-theory approach 19
2 Elements of Dispersion Relation Technique for Two-Body Scattering Reactions 25
2.1 Analytical properties of four-point amplitudes 25
2.1.1 Mandelstam planes for four-point amplitudes 26
2.1.2 Bethe-Salpeter equations in the momentum representation 29
2.2 Dispersion relation N/D-method and ansatz of separable interactions 34
2.2.1 N/D-method for the one-channel scattering amplitude of spinless particles 34
2.2.2 Scattering amplitude and energy non-conservation in the spectral integral representation 37
2.2.3 Composite system wave function and its form factors 39
2.2.4 Scattering amplitude with multivertex representation of separable interaction 42
2.3 Instantaneous interaction and spectral integral equation for two-body systems 45
2.3.1 Instantaneous interaction 45
2.3.2 Spectral integral equation for a composite system 47
2.4 Appendix A. Angular momentum operators 52
2.4.1 Projection operators and denominators of the boson propagators 55
2.4.2 Useful relations for Zαμ1…μn and X(n-1)v2…vn 56
2.5 Appendix B: The ππ scattering amplitude near the two-pion thresholds, π+π- and π0π0 58
2.6 Appendix C: Four-pole fit of the π π (00++) wave in the region Mπ π < 900 MeV 59
3 Spectral Integral Equation for the Decay of a Spinless Particle 63
3.1 Three-body system in terms of separable interactions: analytic continuation of the four-point scattering amplitude to the decay region 64
3.1.1 Final state two-particle S-wave interactions 65
3.1.2 General case: rescatterings of outgoing particles, PiPj → PiPj with arbitrary angular momenta 71
3.2 Non-relativistic approach and transition of two-particle spectral integral to the three-particle one 77
3.2.1 Non-relativistic approach 78
3.2.2 Threshold limit constraint 80
3.2.3 Transition of the two-particle spectral integral representation amplitude to the three-particle spectral integral 81
3.3 Consideration of amplitudes in terms of a three-particle spectral integral 84
3.3.1 Kinematics of the outgoing particles in the c.m. system 86
3.3.2 Calculation of the block B(0)13-12(s, s12) 86
3.4 Three-particle composite systems, their wave functions and form factors 87
3.4.1 Vertex and wave function 88
3.4.2 Three particle composite system form factor 89
3.5 Equation for an amplitude in the case of instantaneous interactions in the final state 90
3.6 Conclusion 91
3.7 Appendix A. Example: loop diagram with GL = GR=1 92
3.8 Appendix B. Phase space for n-particle state 92
3.9 Appendix C. Feynman diagram technique and evolution of systems in the positive time-direction 93
3.9.1 The Feynman diagram technique and non-relativistic three particle systems 94
3.10 Appendix D. Coordinate representation for non-relativistic three-particle wave function 97
4 Non-relativistic Three-Body Amplitude 101
4.1 Introduction 101
4.1.1 Kinematics 101
4.1.2 Basic principles for selecting the diagrams 103
4.2 Non-resonance interaction of the produced particles 106
4.2.1 The structure of the amplitude with a total angular momentum J = 0 106
4.2.2 Production of three particles in a state with J = 1 120
4.3 The production of three particles near the threshold when two particles interact strongly 123
4.3.1 The production of three spinless particles 124
4.4 Decay amplitude for K → 3π and pion interaction 126
4.4.1 The dispersion relation for the decay amplitude 126
4.4.2 Pion spectra and decay ratios in K → π π π within taking into account mass differences of kaons and pions 130
4.4.3 Transformation of the dispersion relation for the K → π π π amplitude to a single integral equation 134
4.5 Equation for the three-nucleon amplitude 137
4.5.1 Method of extraction of the leading singularities 138
4.5.2 Helium-3/tritium wave function 145
4.6 Appendix A. Landau rules for finding the singularities of the diagram 150
4.7 Appendix B. Anomalous thresholds and final state interaction 153
4.8 Appendix C. Homogeneous Skornyakov-Ter-Martirosyan equation 158
4.9 Appendix D. Coordinates and observables in the three-body problem 159
4.9.1 Choice of coordinates and group theory properties 159
4.9.2 Parametrization of a complex sphere 162
4.9.3 The Laplace operator 163
4.9.4 Calculation of the generators Lik and Bik 165
4.9.5 The cubic operator Ω 168
4.9.6 Solution of the eigenvalue problem 170
5 Propagators of Spin Particles and Relativistic Spectral Integral Equations 173
5.1 Boson propagators 174
5.1.1 Projection operators and denominators of the boson propagators 174
5.2 Propagators of fermions 176
5.2.1 The classification of the baryon states 176
5.2.2 Spin-1/2 wave functions 177
5.2.3 Spin-3/2 wave functions 179
5.3 Spectral integral equations for the coupled three-meson decay channels in pp (JPC = 0-+) annihilation at rest 182
5.3.1 The S-P-D-wave meson rescatterings 183
5.3.2 Equations with inclusion of resonance production 188
5.3.3 The coupled decay channels pp(IJPC = 10-+) → π0π0π0, ηηπ0, KKπ0 189
5.4 Conclusion 194
6 Isobar model and partial wave analysis. D-matrix method 197
6.1 The it-Matrix and D-Matrix Techniques 198
6.1.1 K-matrix approach 198
6.1.2 Spectral integral equation for the K-matrix amplitude 200
6.1.3 D-matrix approach 201
6.2 Meson-meson scattering 204
6.2.1 K-matrix fit 204
6.2.2 D-matrix fit 207
6.3 Partial wave analysis of baryon spectra in the frameworks of K-matrix and D-matrix methods 209
6.3.1 Pion and photo induced reactions 212
7 Reggeon-Exchange Technique 217
7.1 Introduction 217
7.2 Meson-nucleon collisions at high energies: peripheral two-meson production in terms of reggeon exchanges 219
7.2.1 K-matrix and D-matrix approaches 220
7.2.2 Reggeized pion-exchange trajectories for the waves JPC = 0++, 1--, 2++, 3--, 4++ 223
7.2.3 Amplitudes with aJ-trajectory exchanges 227
7.2.4 π-p → KK n reaction with exchange by ρ-meson trajectories 233
7.3 Results of the fit 237
7.3.1 The f0(1300) state 239
7.4 Summary for isoscalar resonances 243
7.4.1 Isoscalar-scalar sector 243
7.4.2 Isoscalar-tensor sector 244
7.4.3 Isoscalar sector JPC = 4++ 244
7.5 Appendix A. D-matrix technique in the two-meson production reactions 245
7.5.1 D-matrix in one-channel and two-channel cases 245
7.6 Appendix B. Elements of the reggeon exchange technique in the two-meson production reactions 247
7.6.1 Angular momentum operators for two-meson systems 248
7.6.2 Reggeized pion exchanges 249
7.7 Appendix C. Cross sections for the reactions πN → π πN, KKN, ηηN 256
7.7.1 The CERN-Munich approach 257
7.7.2 GAMS, VES, and BNL approaches 258
7.8 Appendix D. Status of trajectories on (J, M2) plane 259
7.8.1 Kaon trajectories on (J, M2) plane 261
7.9 Appendix E. Assignment of Mesons to Nonets 262
8 Searching for the Quark-Diquark Systematics of Baryons 267
8.1 Diquarks and reduction of baryon states 267
8.2 Baryons as quark-diquark systems 271
8.2.1 S-wave diquarks and baryons 271
8.2.2 Wave functions of quark-diquark systems with L = 0 272
8.2.3 Wave functions of quark-diquark systems with L ≠ 0 277
8.2.4 Quarks and diquarks in baryons 279
8.3 The setting of states with L = 0 and the SU(6) symmetry 282
8.3.1 Baryon spectra for the excited states 283
8.3.2 The setting of (L=0) states 284
8.4 The setting of baryons with L > 0 as (qD11, qD00) states 285
8.4.1 The setting of N(J+) states at MD00 ≠ MD11 and L≥2 288
8.4.2 The setting of the N(J-) states at MD00 ≠ MD11 289
8.4.3 The setting of Δ(J+) states at MD00 ≠ MD11 and L≥2 290
8.4.4 The setting of Δ(J-) states at MD00 ≠ MD11 291
8.4.5 Overlapping of baryon resonances 292
8.5 Version with MD00 = MD11 and overlapping qD00(S = 1/2) and qD11(S = 1/2) states 296
8.6 Conclusion 297
8.7 Appendix A. Spectral integral equations for pure qD00 and qD11 systems 299
8.7.1 Confinement singularities 299
8.7.2 The qD00 systems 300
8.7.3 Spectral integral equations for qD11 systems with I = 3/2 303
8.8 Appendix B. Group theoretical description. Symmetrical basis in the three-body problem 307
8.8.1 Group theoretical properties. Parametrization 309
8.8.2 Basis functions 311
8.8.3 Transformation coefficients (j'1j'2|j'1j2)φKJM 313
8.8.4 Applying (j'1j'2|j'1j2)φKLM the three-body problem 318
8.8.5 The d-function of the O(6) group 319
9 Conclusion 323
