Contents -- Preface -- References -- 1. Introduction -- 1.1 Non-relativistic three-nucleon and three-quark systems -- 1.1.1 Description of three-nucleon systems -- 1.1.2 Three-quark systems -- 1.2 Dispersion relation technique for three particle systems -- 1.2.1 Elements of the dispersion relation technique for two-particle systems -- 1.2.2 Interconnection of three particle decay amplitudes and two-particle scattering ones in hadron physics -- 1.2.3 Quark-gluon language for processes in regions I, III and IV -- 1.2.4 Spectral integral equation for three particles -- 1.2.5 Isobar models -- 1.2.5.1 Amplitude poles -- 1.2.5.2 D-matrix propagator for an unstable particle and the K matrix amplitude -- 1.2.5.3 K-matrix and D-matrix masses and the amplitude pole -- 1.2.5.4 Accumulation of widths of overlapping resonances -- 1.2.5.5 Loop diagrams with resonances in the intermediate states -- 1.2.5.6 Isobar model for high energy peripheral production processes -- 1.2.6 Quark-diquark model for baryons and group theory approach -- 1.2.6.1 Quark-diquark model for baryons -- References -- 2. Elements of Dispersion Relation Technique for Two-Body Scattering Reactions -- 2.1 Analytical properties of four-point amplitudes -- 2.1.1 Mandelstam planes for four-point amplitudes -- 2.1.1.1 Dalitz plot for the 4 → 1 + 2 + 3 decay -- 2.1.2 Bethe-Salpeter equations in the momentum representation -- 2.1.2.1 Multiparticle intermediate states -- 2.1.2.2 Composite systems -- 2.1.2.3 Zoo-diagrams in the BS-equation -- 2.1.2.4 Miniconclusion: two-particle composite systemsin the BS-equation -- 2.2 Dispersion relation N/D-method and ansatz of separable interactions -- 2.2.1 N/D-method for the one-channel scattering amplitude of spinless particles -- 2.2.2 Scattering amplitude and energy non-conservation in the spectral integral representation.

2.2.3 Composite system wave function and its form factors -- 2.2.4 Scattering amplitude with multivertex representation of separable interaction -- 2.2.4.1 Generalization for an arbitrary angular momentum state, L = J -- 2.3 Instantaneous interaction and spectral integral equation for two-body systems -- 2.3.1 Instantaneous interaction -- 2.3.1.1 Coordinate representation -- 2.3.1.2 Instantaneous interaction - transformation into a set of separable vertices -- 2.3.1.3 An example: expressions for q⊥ and t⊥ in the centre-of mass system -- 2.3.2 Spectral integral equation for a composite system -- 2.3.2.1 Spectral integral equation for vertex function with L = 0 -- 2.3.2.2 Spectral integral equation for the (L = 0)-wave function -- 2.3.2.3 The spectral integral equation for the states with angular momentum L -- 2.4 Appendix A. Angular momentum operators -- 2.4.1 Projection operators and denominators of the boson propagators -- 2.4.2 Useful relations for Z μ1...μn and X(n−1)ν2...νn -- 2.5 Appendix B: The ππ scattering amplitude near the two-pion thresholds, + − and 0 0 -- 2.6 Appendix C: Four-pole fit of the (00++) wave in the region M < 900 MeV -- References -- 3. Spectral Integral Equation for the Decay of a Spinless Particle -- 3.1 Three-body system in terms of separable interactions: analytic continuation of the four-point scattering amplitude to the decay region -- 3.1.1 Final state two-particle S-wave interactions -- 3.1.1.1 Calculation of A(0)12 (s12) in the c.m.s. of particles P1P2 -- 3.1.1.2 Analytic continuation of the integration contour over z -- 3.1.1.3 Threshold behavior of the three-particle amplitude A(Jin=0)P1P2P3(s12, s13, s23) -- 3.1.1.4 Spectral integral equations for S-wave interactions -- 3.1.2 General case: rescatterings of outgoing particles, PiPj → PiPj , with arbitrary angular momenta.

3.1.2.1 Decay (JPin = 0−)-state → P1P2P3 -- 3.1.2.2 Miniconclusion -- 3.2 Non-relativistic approach and transition of two-particlespectral integral to the three-particle one -- 3.2.1 Non-relativistic approach -- 3.2.2 Threshold limit constraint -- 3.2.3 Transition of the two-particle spectral integral representation amplitude to the three-particle spectral integral -- 3.3 Consideration of amplitudes in terms of a three-particlespectral integral -- 3.3.1 Kinematics of the outgoing particles in the c.m. system -- 3.3.2 Calculation of the block B(0)13−12(s, s12) -- 3.4 Three-particle composite systems, their wave functions and form factors -- 3.4.1 Vertex and wave function -- 3.4.2 Three particle composite system form factor -- 3.5 Equation for an amplitude in the case of instantaneous interactions in the final state -- 3.6 Conclusion -- 3.7 Appendix A. Example: loop diagram with GL=GR=1 -- 3.8 Appendix B. Phase space for n-particle state -- 3.9 Appendix C. Feynman diagram technique and evolution of systems in the positive time-direction -- 3.9.1 The Feynman diagram technique and non relativistic three particle systems -- 3.9.1.1 Two particle c.m. system, k1 + k2 = 0 -- 3.9.1.2 Three particle c.m. system, k1 + k2 + k3 = 0 -- 3.10 Appendix D. Coordinate representation for non-relativistic three-particle wave function -- References -- 4. Non-relativistic Three-Body Amplitude -- 4.1 Introduction -- 4.1.1 Kinematics -- 4.1.2 Basic principles for selecting the diagrams -- 4.2 Non-resonance interaction of the produced particles -- 4.2.1 The structure of the amplitude with a total angular momentum J = 0 -- 4.2.1.1 Terms of the order of ∼ const and ∼

4.2.1.5 Amplitude and total cross section up to terms ∼ E -- 4.2.1.6 Miniconclusion -- 4.2.2 Production of three particles in a state with J = 1 -- 4.3 The production of three particles near the threshold when two particles interact strongly -- 4.3.1 The production of three spinless particles -- 4.4 Decay amplitude for K → 3 and pion interaction -- 4.4.1 The dispersion relation for the decay amplitude -- 4.4.2 Pion spectra and decay ratios in K → within taking into account mass differences of kaons and pions -- 4.4.2.1 Decay ratios in K → -- 4.4.2.2 Cusps in pion spectra at small relative momenta -- 4.4.3 Transformation of the dispersion relation for the K → amplitude to a single integral equation -- 4.5 Equation for the three-nucleon amplitude -- 4.5.1 Method of extraction of the leading singularities -- 4.5.1.1 Amplitude of production of three spinless particles -- 4.5.1.2 Separation of the threshold singularities and the cutoff procedure -- 4.5.1.3 Extraction of the leading singularities -- 4.5.2 Helium-3/tritium wave function -- 4.5.2.1 Miniconclusion -- 4.6 Appendix A. Landau rules for finding the singularities of the diagram -- 4.7 Appendix B. Anomalous thresholds and final state interaction -- 4.8 Appendix C. Homogeneous Skornyakov-Ter-Martirosyan equation -- 4.9 Appendix D. Coordinates and observables in the three body problem -- 4.9.1 Choice of coordinates and group theory properties -- 4.9.2 Parametrization of a complex sphere -- 4.9.3 The Laplace operator -- 4.9.4 Calculation of the generators Lik and Bik -- 4.9.5 The cubic operator Ω -- 4.9.6 Solution of the eigenvalue problem -- References -- 5. Propagators of Spin Particles and Relativistic Spectral Integral Equations -- 5.1 Boson propagators -- 5.1.1 Projection operators and denominators of the boson propagators -- 5.1.1.1 The photon projection operator.

5.2 Propagators of fermions -- 5.2.1 The classification of the baryon states -- 5.2.2 Spin-1/2 wave functions -- 5.2.3 Spin-3/2 wave functions -- 5.2.3.1 Wave function for Δ -- 5.2.3.2 Wave function for Δ -- 5.2.3.3 Baryon projection operators -- 5.2.3.4 Projection operators for particles with J > 1/2 -- 5.3 Spectral integral equations for the coupled three-meson decaychannels in pp (JPC = 0−+) annihilation at rest -- 5.3.1 The S-P-D-wave meson rescatterings -- 5.3.1.1 P-wave interaction in the final state -- 5.3.1.2 D-wave interaction in the final state -- 5.3.2 Equations with inclusion of resonance production -- 5.3.2.1 Two-particle scattering amplitude -- 5.3.2.2 Three-particle production amplitude -- 5.3.3 The coupled decay channels pp(IJPC = 10−+) →π0π0π0, ηηπ0, KKπ0 -- 5.3.3.1 Reaction pp (10−+) → 0 0 0 -- 5.3.3.2 Reaction pp(0−+) → ηη 0 -- 5.3.3.3 Reaction pp(0−+) → K K 0 -- 5.4 Conclusion -- References -- 6. Isobar model and partial wave analysis. D-matrix method -- 6.1 The K-Matrix and D-Matrix Techniques -- 6.1.1 K-matrix approach -- 6.1.2 Spectral integral equation for the K-matrix amplitude -- 6.1.3 D-matrix approach -- 6.2 Meson-meson scattering -- 6.2.1 K-matrix fit -- 6.2.2 D-matrix fit -- 6.3 Partial wave analysis of baryon spectra in the frameworks of K-matrix and D-matrix methods -- 6.3.1 Pion and photo induced reactions -- References -- 7. Reggeon-Exchange Technique -- 7.1 Introduction -- 7.2 Meson-nucleon collisions at high energies: peripheral two meson production in terms of reggeon exchanges -- 7.2.1 K-matrix and D-matrix approaches -- 7.2.1.1 K-matrix approach -- 7.2.1.2 D-matrix approach -- 7.2.2 Reggeized pion-exchange trajectories for the waves JPC = 0++, 1−−, 2++, 3−−, 4++ -- 7.2.2.1 Kinematics for reggeon exchange amplitudes -- 7.2.2.2 Amplitude with leading and daughter pion trajectory exchanges.

7.2.2.3 The t-channel 2 exchange.