Vector Meson Production and Nucleon Resonance Analysis in a Coupled-Channel Approach Inaugural-Dissertation zur Erlangung des Doktorgrades der Naturwissenschaftlichen Fakult¨at der Justus-Liebig-Universit¨atGießen Fachbereich 07 – Mathematik, Physik, Geographie vorgelegt von Gregor Penner aus Gießen Gießen, 2002 D 26 Dekan: Prof. Dr. Albrecht Beutelspacher I. Berichterstatter: Prof. Dr. Ulrich Mosel II. Berichterstatter: Prof. Dr. Volker Metag Tag der m¨undlichen Pr¨ufung: Contents 1 Introduction 1 2 The Bethe-Salpeter Equation and the K-Matrix Approximation 7 2.1 Bethe-Salpeter Equation ........................................ 7 2.2 Unitarity and the K-Matrix Approximation ......................... 11 3 The Model 13 3.1 Other Models Analyzing Pion- and Photon-Induced Reactions on the Nucleon 15 3.1.1 Resonance Models: ...................................... 15 3.1.2 Separable Potential Models ................................ 17 3.1.3 Effective Lagrangian Models ............................... 17 3.2 The Giessen Model ............................................ 20 3.3 Asymptotic Particle (Born) Contributions .......................... 23 3.3.1 Electromagnetic Interactions ............................... 23 3.3.2 Hadronic Interactions .................................... 25 3.4 Baryon Resonances ............................................ 28 3.4.1 (Pseudo-)Scalar Meson Decay .............................. 28 3.4.2 Electromagnetic Decays .................................. 32 3.4.3 Vector Meson Decays .................................... 33 3.5 Intermediate Mesons ........................................... 33 3.5.1 Electromagnetic Decays .................................. 34 3.5.2 Hadronic Decays ........................................ 34 3.6 Formfactors ................................................. 35 3.7 Formfactors, Photons, and Gauge Invariance ........................ 37 I 3.7.1 Born Contributions and Electromagnetic Formfactors ........... 37 3.7.2 Born Contributions and Hadronic Formfactors ................. 37 3.7.3 Resonances and Intermediate Mesons ........................ 39 3.7.4 Formfactors and Free Parameters ........................... 40 3.8 Model Parameters ............................................ 40 4 Helicity Amplitudes and Partial-Wave Decomposition of the BS Equa- tion 45 4.1 Helicity Amplitudes ........................................... 46 4.2 Partial-Wave Decomposition of the BS Equation ..................... 49 5 Calculation of Amplitudes 51 1 5.1 Spin-0 Spin- 2 Scattering ........................................ 52 5.1.1 Mesons with Identical Parity ............................... 52 5.1.2 Mesons with Opposite Parity .............................. 53 5.2 Pion-Induced Vector Meson Production ............................ 53 5.2.1 Pseudoscalar (J P = 0¡) Mesons ............................ 54 5.2.2 Scalar (J P = 0+) Mesons ................................. 55 5.3 Vector Meson Nucleon Scattering ................................. 56 5.4 Reactions Including Photons: Gauge Invariance ...................... 60 5.4.1 Photoproduction of (Pseudo-) Scalar Mesons .................. 60 5.4.2 Photoproduction of Vector Mesons .......................... 63 5.4.3 Compton Scattering ..................................... 63 6 Partial Waves and Multipoles 65 6.1 ¼N Scattering ................................................ 65 6.2 Photoproduction .............................................. 66 6.2.1 Photoproduction of Pions ................................. 69 6.2.2 Compton Scattering ..................................... 70 II 7 Experimental Data and Fitting Strategy 73 7.1 Partial-Wave Analyses ......................................... 74 7.1.1 Pion Production ........................................ 74 7.1.2 Two-Pion Production .................................... 77 7.2 Direct Comparison with Observables .............................. 79 7.2.1 γN ! γN ............................................. 80 7.2.2 γN ! ´N ............................................. 80 7.2.3 γN ! KΛ ............................................. 81 7.2.4 γN ! KΣ ............................................. 81 7.2.5 γN ! !N ............................................. 82 7.2.6 ¼N ! ´N ............................................. 82 7.2.7 ¼N ! KΛ ............................................. 83 7.2.8 ¼N ! KΣ ............................................. 83 7.2.9 ¼N ! !N ............................................. 84 7.3 Data-Base Balance and Fitting Strategy ........................... 91 7.4 Fitting Strategy .............................................. 93 8 Results 97 8.1 Results on Pion-Induced Reactions ............................... 99 3 8.1.1 σ meson, Chiral Symmetry, and Spin- 2 Vertices ................ 101 8.1.2 ¼N ! ¼N ............................................. 103 8.1.3 ¼N ! 2¼N ............................................ 106 8.1.4 ¼N ! ´N ............................................. 110 8.1.5 ¼N ! KΛ ............................................. 114 8.1.6 ¼N ! KΣ ............................................. 116 8.1.7 ¼N ! !N ............................................. 122 8.2 Extracted Hadronic Parameters .................................. 124 8.2.1 Background Contributions and t-Channel Formfactors ........... 124 8.2.2 Scattering Lengths ...................................... 129 8.2.3 Resonance Masses and Widths ............................. 130 8.3 Summary of Pion-Induced Results ................................ 142 III 8.4 Results on Photon-Induced Reactions ............................. 144 8.4.1 Compton Scattering ..................................... 144 8.4.2 Pion Photoproduction .................................... 147 8.4.3 ´ Photoproduction ...................................... 153 8.4.4 KΛ Photoproduction .................................... 157 8.4.5 KΣ Photoproduction .................................... 160 8.4.6 ! Photoproduction ...................................... 164 8.4.7 Photoabsorption on the Nucleon ............................ 166 8.4.8 Resonance Electromagnetic Helicity Amplitudes ................ 168 8.5 Summary of Photoproduction Results ............................. 172 9 Summary and Outlook 175 A Notation and Normalization 179 A.1 Metric and Momentum States ................................... 179 A.2 Dirac Matrices, Spinors, and all the Rest ........................... 180 1 A.2.1 Dirac (Spin- 2 ) Spinors ................................... 182 A.2.2 Spin-1 Polarization Vectors ................................ 183 3 A.2.3 Rarita-Schwinger (Spin- 2 ) Spinors .......................... 185 B Legendre Polynomials and Wigner Simplified Rotation Matrices 187 B.1 Legendre Polynomials .......................................... 187 B.2 Wigner Simplified Rotation Matrices (d-functions) ................... 187 C Formal Scattering Theory 190 C.1 Fundamental Matrices ......................................... 190 C.2 Optical Theorem ............................................. 190 C.3 Matrix Relations and Notations .................................. 191 D Properties of the Bethe-Salpeter Equation 193 D.1 Connection to the Optical Theorem ............................... 193 D.2 Imaginary Part of the Bethe-Salpeter Propagator .................... 194 IV E Lagrangians, Widths, and Couplings 197 E.1 Born and t-channel Interactions .................................. 197 E.2 Baryon Resonance Interactions ................................... 201 1 E.2.1 Spin- 2 Resonances ....................................... 201 3 E.2.2 Spin- 2 Resonances ....................................... 203 F Isospin 205 F.1 Isospin Decomposed Amplitudes ................................. 206 F.1.1 Hadronic reactions ...................................... 206 F.1.2 Reactions Involving Photons ............................... 209 F.2 Isospin Operators in the Interaction Lagrangians ..................... 212 F.2.1 Isospin Decomposed Photon Couplings ....................... 212 F.2.2 Isospin Decomposed Photons and Gauge Invariance ............. 215 G Observables and Partial Waves 219 G.1 Cross Sections ................................................ 219 G.2 Polarization Observables ....................................... 220 G.2.1 Pion-Induced (Pseudo-) Scalar Meson Production .............. 221 G.2.2 Photoproduction of (Pseudo-) Scalar Mesons .................. 221 G.2.3 Compton Scattering ..................................... 222 G.2.4 Photoproduction of Vector Mesons .......................... 222 H Parameters of Further Calculations 223 I Numerical Methods and Extraction of Partial Waves 230 I.1 Extraction of Feynman Diagram Contributions and Calculation of Observ- ables ....................................................... 230 I.2 Performance Optimization ...................................... 232 Bibliography 235 Deutsche Zusammenfassung 253 Danksagung 259 Chapter 1 Introduction “Nucleons are the stuff of which our world is made. As such they must be at the center of any discussion of why the world we actually experience has the character it does.” Nathan Isgur, Why N ¤’s are Important, overview talk at N¤2000 Conference, Thomas Jefferson Laboratory, Newport News, VA/USA, 16-19 Febr. 2000. “One very powerful way of experimentally investigating the strongly in- teracting particles (hadrons) is to look at them, to probe them with a known particle; in particular the photon (no other is known as well). This permits a much finer control of variables, and probably decreases the theoretical complexity of the interactions.”