THE THEORY OF COMPLEX ANGULAR MOMENTA
CONTENTS HIGH ENERGY HADRON SCATTERING BASIC PRINCIPLES 1 1 1 INVARIANT SCATTERING AMPLITUDE AND CROSS SECTION 1 1 2 ANALYTICITY AND CAUSALITY 1 1 3 SINGULARITIES 1 1 4 CROSSING SYMMETRY 1 1 5 THE UNITARITY CONDITION FOR THE SCATTERING MATRIX MANDELSTAM VARIABLES FOR TWO-PARTICLE SCATTERING 1 2 1 THE MANDELSTAM PLANE 1 2 2 THRESHOLD SINGULARITIES ON THE MANDELSTAM PLANE PARTIAL WAVE EXPANSION AND UNITARITY 1 3 1 THRESHOLD BEHAVIOUR OF PARTIAL WAVE AMPLITUDES 1 3 2 SINGULARITIES OF IM A ON THE MANDELSTAM PLANE (KARPLUS CURVE) THE FROISSART THEOREM THE POMERANCHUK THEOREM PHYSICS OF THE T-CHANNEL AND COMPLEX ANGULAR MOMENTA ANALYTICAL CONTINUATION OF THE T-CHANNEL UNITARITY CONDITION 2 1 1 THE MANDELSTAM REPRESENTATION 2 1 2 INCONSISTENCY OF THE ‘BLACK DI’ MODEL OF DI?RACTION COMPLEX ANGULAR MOMENTA PARTIAL WAVE EXPANSION AND SOMMERFELD–WATSON REPRESENTATION CONTINUATION OF PARTIAL WAVE AMPLITUDES TO COMPLEX ? 2 4 1 NON-RELATIVISTIC QUANTUM MECHANICS 2 4 2 RELATIVISTIC THEORY GRIBOV–FROISSART PROJECTION T-CHANNEL PARTIAL WAVES AND THE BLACK DISK MODEL SINGULARITIES OF PARTIAL WAVES AND UNITARITY CONTINUATION OF PARTIAL WAVES WITH COMPLEX ?TO T < 3 1 1 THRESHOLD SINGULARITY AND PARTIAL WAVES ?? 3 1 2 ??(T) AT T < 0 AND ITS DISCONTINUITY THE UNITARITY CONDITION FOR PARTIAL WAVES WITH COMPLEX ? SINGULARITIES OF THE PARTIAL WAVE AMPLITUDE 3 3 1 LEFT CUT IN NON-RELATIVISTIC THEORY 3 3 2 FIXED SINGULARITIES 3 3 3 MOVING SINGULARITIES MOVING POLES AND RESONANCES PROPERTIES OF REGGE POLES RESONANCES BOUND STATES ELEMENTARY PARTICLE OR BOUND STATE? 4 3 1 REGGE TRAJECTORIES 4 3 2 REGGE POLE EXCHANGE AND PARTICLE EXCHANGE (T > 0) 4 3 3 REGGE EXCHANGE AND ELEMENTARY PARTICLES (T < 0) 4 3 4 THERE IS NO ELEMENTARY PARTICLE WITH J > 4 3 5 ASYMPTOTICS OF S-CHANNEL AMPLITUDES AND REGGEIZATION FACTORIZATION REGGE POLES IN HIGH ENERGY SCATTERING T-CHANNEL DOMINANCE ELASTIC SCATTERING AND THE POMERON 5 2 1 QUANTUM NUMBERS OF THE POMERON 5 2 2 SLOPE OF THE POMERON TRAJECTORY SHRINKAGE OF THE DI?RACTIVE CONE 5 3 1 S-CHANNEL PARTIAL WAVES IN THE IMPACT PARAMETER SPACE RELATION BETWEEN TOTAL CROSS SECTIONS SCATTERING OF PARTICLES WITH SPIN VECTOR PARTICLE EXCHANGE SCATTERING OF NUCLEONS 6 2 1 REGGEON QUANTUM NUMBERS AND N ¯N ?REGGEON VERTICES 6 2 2 VACUUM POLE IN ?N AND NN SCATTERING CONSPIRACY FERMION REGGE POLES BACKWARD SCATTERING AS A RELATIVISTIC E?ECT PION–NUCLEON SCATTERING 7 2 1 PARITY IN THE U-CHANNEL 7 2 2 FERMION POLES WITH DE?NITE PARITY AND SINGULARITY AT U = 7 2 3 OSCILLATIONS IN THE FERMION POLE AMPLITUDE REGGEIZATION OF A NEUTRON REGGE POLES IN PERTURBATION THEORY SCATTERING OF A PARTICLE IN AN EXTERNAL ?ELD 8 2 1 G?3 THEORY IN THE DU?N–KEMMER FORMALISM 8 2 2 ANALYTIC PROPERTIES OF THE AMPLITUDES 8 2 3 ORDER G4 LN S 8 2 4 ORDER G6 LN2 S 8 2 5 LADDER DIAGRAMS IN ALL ORDERS 8 2 6 NON-LADDER DIAGRAMS INTERACTION WITH VECTOR MESONS REGGEIZATION OF AN ELECTRON COMPTON SCATTERING AMPLITUDE ELECTRON REGGE POLES 9 2 1 CONSPIRACY IN PERTURBATION THEORY 9 2 2 REGGEIZATION IN QED (WITH MASSLESS PHOTON) ELECTRON REGGEIZATION: NONSENSE STATES VECTOR ?ELD THEORY RˆOLE OF SPIN E?ECTS IN REGGEIZATION 10 1 1 NONSENSE STATES IN THE UNITARITY CONDITION 10 1 2 ITERATION OF THE UNITARITY CONDITION 10 1 3 NONSENSE STATES FROM THE S- AND T-CHANNEL POINTS OF VIEW 10 1 4 THE J = 2 POLE IN THE PERTURBATIVE NONSENSE–NONSENSE AMPLITUDE QED PROCESSES WITH PHOTONS IN THE T-CHANNEL 10 2 1 THE VACUUM CHANNEL IN QED 10 2 2 THE PROBLEM OF THE PHOTON REGGEIZATION INCONSISTENCY OF THE REGGE POLE PICTURE THE POLE ?= ?1 AND RESTRICTION ON THE AMPLITUDE FALL-O? CONTRADICTION WITH UNITARITY 11 3 1 AMPLITUDE CANNOT FALL FASTER THAN 1/S PARTICLES WITH SPIN: FAILURE OF THE REGGE POLE PICTURE TWO-REGGEON EXCHANGE AND BRANCH POINT SINGULARITIES IN THE ?PLANE NORMALIZATION OF PARTIAL WAVES AND THE UNITARITY CONDITION 12 1 1 REDE?NITION OF PARTIAL WAVE AMPLITUDES 12 1 2 PARTICLES WITH SPIN IN THE UNITARITY CONDITION PARTICLE SCATTERING VIA A TWO-PARTICLE INTERMEDIATE STATE TWO-REGGEON EXCHANGE AND PRODUCTION VERTICES ASYMPTOTICS OF TWO-REGGEON EXCHANGE AMPLITUDE MOVEMENT OF THE BRANCHING IN THE T AND J PLANES SIGNATURE OF THE TWO-REGGEON BRANCHING PROPERTIES OF MANDELSTAM BRANCH SINGULARITIES BRANCHINGS AS A GENERALIZATION OF THE ?= ?1 SINGULARITY 13 1 1 BRANCHINGS IN THE J PLANE 13 1 2 BRANCH SINGULARITY IN THE UNITARITY CONDITION BRANCHINGS IN THE VACUUM CHANNEL 13 2 1 THE PATTERN OF BRANCH POINTS IN THE J PLANE 13 2 2 THE MANDELSTAM REPRESENTATION IN THE PRESENCE OF BRANCHINGS VACUUM–NON-VACUUM POLE BRANCHINGS EXPERIMENTAL VERI?CATION OF BRANCHING SINGULARITIES 13 4 1 BRANCHINGS AND CONSPIRACY REGGEON DIAGRAMS TWO-PARTICLE–TWO-REGGEON TRANSITION AMPLITUDE 14 1 1 STRUCTURE OF THE VERTEX 14 1 2 ANALYTIC PROPERTIES OF THE VERTEX 14 1 3 FACTORIZATION PARTIAL WAVE AMPLITUDE OF THE MANDELSTAM BRANCHING INTERACTING REGGEONS REGGEON ?ELD THEORY ENHANCED REGGEON DIAGRAMS E?ECTIVE ?ELD THEORY OF INTERACTING REGGEONS EQUATION FOR THE GREEN FUNCTION G WEAK AND STRONG COUPLING REGIMES POMERON GREEN FUNCTION AND REGGEON UNITARITY CONDITION THE STRUCTURE OF WEAK AND STRONG COUPLING SOLUTIONS WEAK COUPLING REGIME 17 1 1 THE GREEN FUNCTION 17 1 2 P ?PP VERTEX 17 1 3 INDUCED MULTI-REGGEON VERTICES 17 1 4 VANISHING OF MULTI-REGGEON COUPLINGS PROBLEMS OF THE STRONG COUPLING REGIME APPENDIX A: SPACE-TIME DESCRIPTION OF THE HADRON INTERACTIONS AT HIGH ENERGIES WAVE FUNCTION OF THE HADRON ORTHOGONALITY AND NORMALIZATION DISTRIBUTION OF THE PARTONS IN SPACE AND MOMENTUM DEEP INELASTIC SCATTERING STRONG INTERACTIONS OF HADRONS ELASTIC AND QUASI-ELASTIC PROCESSES REFERENCES APPENDIX B: CHARACTER OF INCLUSIVE SPECTRA AND ?UCTUATIONS PRODUCED IN INELASTIC PROCESSES BY MULTI-POMERON EXCHANGE THE ABSORPTIVE PARTS OF REGGEON DIAGRAMS IN THE S-CHANNEL CLASSI?CATION OF INELASTIC PROCESSES RELATIONS AMONG THE ABSORPTIVE PARTS OF REGGEON DIAGRAMS INCLUSIVE CROSS SECTIONS MAIN CORRECTIONS TO THE INCLUSIVE CROSS SECTIONS IN THE CENTRAL REGION FLUCTUATIONS IN THE DISTRIBUTION OF THE DENSITY OF PRODUCED PARTICLES APPENDIX C: THEORY OF THE HEAVY POMERON INTRODUCTION NON-ENHANCED CUTS AT ?? = ESTIMATION OF ENHANCED CUTS AT ?? = STRUCTURE OF THE TRANSITION AMPLITUDE OF ONE POMERON - TO TWO THE GREEN FUNCTION AND THE VERTEX PART AT ?? = PROPERTIES OF HIGH ENERGY PROCESSES IN THE THEORY WITH ?? = C 6 1 TWO-PARTICLE PROCESSES, TOTAL CROSS SECTIONS C 6 2 INCLUSIVE SPECTRA, MULTIPLICITY C 6 3 CORRELATION, MULTIPLICITY DISTRIBUTION C 6 4 PROBABILITY OF ?UCTUATIONS IN INDIVIDUAL EVENTS: THE INCLUSIVE SPECTRUM IN THE THREE-POMERON LIMIT C 6 5 MULTI-REGGEON PROCESSES THE CASE ?? ?= THE CONTRIBUTION OF CUTS AT SMALL ?? CONCLUSION