Hyperon Interaction in Free Space and Nuclear Matter Within a SU(3) Based Meson Exchange Model Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) Madhumita Dhar aus Cooch-Behar, Indien Institut f¨urTheoretische Physik Justus-Liebig-Universit¨at Gießen, Germany June, 2016 2 Dekan: Prof. Dr. Bernhard M¨uhlherr Betreuer: Prof. Dr. Horst Lenske Erstgutachter: Prof. Dr. Horst Lenske Zweigutachter: Prof. Dr. Christian Fischer To my parents \It always seems impossible until its done." Nelson Mandela Abstract To establish the connection between free space and in-medium hyperon-nucleon in- teractions is the central issue of this thesis. The guiding principle is flavor SU(3) symmetry which is exploited at various levels. In first step hyperon-nucleon and hyperon- hyperon interaction boson exchange potential in free space are introduced. A new parameter set applicable for the complete baryon octet has been derived leading to an updated one-boson- exchange model, utilizing SU(3) flavor symmetry, optimizing the number of free parameters involved, and revising the set of mesons included. The scalar, pseudoscalar, and vector SU(3) meson octets are taken into account. T-matrices are calculated by solving numerically coupled linear systems of Lippmann-Schwinger equations obtained from a 3-D reduced Bethe-Salpeter equa- tion. Coupling constants were determined by χ2 fits to the world set of scattering data. A good description of the few available data is achieved within the imposed SU(3) constraints. Having at hand a consistently derived vacuum interaction we extend the ap- proach next to investigations of the in-medium properties of hyperon interaction, avoiding any further adjustments. Medium effect in infinite nuclear matter are treated microscopically by recalculating T-matrices by an medium-modified system of Lippmann-Schwinger equations. A particular important role is played by the Pauli projector accounting for the exclusion principle. The presence of a background me- dium induces a weakening of the vacuum interaction amplitudes. Especially coupled channel mixing is found to be affected sensitively by medium. Investigation on scat- 5 6 tering lengths and effective range parameters are revealing the density dependence of the interaction on a quantitative level. Abstrakt Der zentrale Aspekt dieser Arbeit ist es die Beziehung zwischen der Hyperon- Nukleon Wechselwirkung im Vakuum und Medium herzustellen. Das Leitprin- zip ist die SU(3) flavour Symmetrie die auf verschiedenen Levels Verwendung fin- det. In einem ersten Schritt werden die Bosonenaustauschpotentiale der Hyperon- Nukleon und Hyperon-Hyperon Wechselwirkung eingef¨uhrt. Ein neuer Paramet- ersatz, welcher f¨urdas gesamte Baryon-Oktett anwendbar ist wurde bestimmt, was unter Benutzung der SU(3) flavour Symmetrie, Optimierung der Anzahl beteiligter freier Parameter und Uberarbeitung¨ des einbezogenen Satzes an Mesonen zu einem aktualisierten Einbosonaustausch-Modell f¨uhrt. Die skalaren, pseudoskalaren und vektoriellen SU(3) Meson-Oktetts sind ber¨ucksichtigt. T-Matrizen sind durch das numerische L¨osengekoppelter, linearer Systeme von Lippmann-Schwinger-Gleichungen, erhalten aus einer dreidimensionalen reduzierten Bethe-Salpeter-Gleichung, berech- net. Kopplungskonstanten wurden durch einen χ2-Fit an den weltweiten Satz an Streudaten bestimmt. Eine gute Beschreibung der wenigen, verf¨ugbarenDaten ist innerhalb der auferlegten SU(3) Bedingungen erreicht. Mit einer konsistent bestimmten Vakuumwechselwirkung zur Hand, erweitern wir den Ansatz, unter Vermeidung irgendwelcher weiterer Anpassungen zu Untersuchun- gen der Mediumseigenschaften der Hyperonwechselwirkung. Mediumseffekte in un- endlicher Kernmaterie sind mikroskopisch durch die Neuberechnung der T-Matrizen durch ein mediumsmodifiziertes System von Lippmann-Schwinger-Gleichungen be- handelt. Eine besonders wichtige Rolle spielt der Pauli-Projektor, welcher das Aus- schlussprinzip ber¨ucksichtigt. Das Vorhandensein eines Hintergrundmediums be- 7 8 wirkt eine Abschw¨achung der Vakuumswechselwirkunsgamplituden. Es stellt sich heraus, dass insbesondere die Mischung gekoppelter Kan¨aledurch das Medium em- pfindlich beeinflusst ist. Die Untersuchungen von Streul¨angenund Parametern der effektiven Reichweite enth¨ullendie quantitative Dichteabh¨angigkeit der Wechsel- wirkung. Contents 1 Introduction 19 1.1 Strong Interaction ............................. 19 1.1.1 Properties of Nuclear Forces ................... 20 1.2 Pathway to Hyperon Interaction ..................... 22 1.3 Current Approaches ............................ 24 1.3.1 Lattice QCD approaches ..................... 24 1.3.2 Meson Exchange Models ..................... 25 1.3.3 Chiral (χ) Effective Field Theory ................ 26 1.3.4 Quark Cluster Model ....................... 26 1.4 Framework Used in This Thesis ..................... 27 2 Interaction Model Description 29 2.1 SUf (3) Flavor Symmetry ......................... 30 2.1.1 Baryon and Meson Representations in SUf (3) ......... 33 2.2 Interaction Lagrangian .......................... 35 2.2.1 Field Theoretical Description .................. 36 2.2.2 Meson Exchange Forces ..................... 36 2.3 Effective Model Parameters ....................... 40 2.3.1 Meson Masses ........................... 40 2.3.2 Coupling Constants ........................ 42 2.3.3 Octet-Octet and Octet-Singlet Interaction ........... 44 2.3.4 SUf (3) Baryon-Baryon-Meson vertices ............. 45 2.3.5 Free Parameters .......................... 48 2.4 Comparison with Contemporary Models ................ 50 2.5 Summary of the model .......................... 52 3 Scattering Theory and Formalism 53 3.1 Kinematics ................................ 54 3.1.1 Relativistic kinematics of Two-particle Scattering ....... 54 3.2 Scattering Equations ........................... 57 3.2.1 Bethe-Salpeter Equation ..................... 57 3.2.2 Lippmann-Schwinger Equation ................. 60 3.3 Isospin Basis ................................ 61 3.4 Coupled Channel Formalism ....................... 64 9 10 CONTENTS 3.5 Formation of the Potential ........................ 68 3.5.1 OBEP Amplitudes ........................ 69 3.5.2 Potential Matrix in Isospin Basis ................ 78 3.6 Introduction to Particle Basis ...................... 79 4 Numerical Methods 85 4.1 R-Matrix ................................. 85 4.2 Representation in Partial Wave Basis .................. 86 4.2.1 Partial Wave Amplitude ..................... 87 4.2.2 Partial Wave Projection of Potential Elements ......... 91 4.2.3 Partial Wave Integral Equation ................. 92 4.3 Numerical Formalism for R-matrix Solution .............. 94 4.3.1 Extraction of T-matrix from R-matrix ............. 97 4.4 Determination of Observables ...................... 99 4.4.1 Cross Section ........................... 99 4.4.2 Phase Shift ............................ 100 4.4.3 Low-Energy (LE) Parameters .................. 101 5 Vacuum Hyperon-Baryon Interactions 105 5.1 Sensitivity of Parameters ........................ 105 5.1.1 Fitting Procedure ......................... 107 5.2 Result of Fit ................................ 116 5.3 Free Space Result ............................. 121 5.3.1 S = 0 Results ........................... 122 5.3.2 S = -1 Results ........................... 126 5.3.2.1 Uncoupled Channels .................. 127 5.3.2.2 Coupled Channels ................... 128 5.3.3 S = -2 Results ........................... 139 5.4 Dependence of the LE Parameters on the Coupling Constants .... 141 5.5 Summary of Free Space Interaction ................... 142 6 In-Medium Effect 147 6.1 Baryon-Baryon Interaction in Infinite Nuclear Matter ......... 148 6.1.1 In-Medium Bethe-Goldstone Equation ............. 148 6.1.2 Pauli Projector Operator ..................... 150 6.1.3 Transformation to Center-of-Mass Frame ............ 151 6.2 Hyperon Mean Fields ........................... 155 6.3 In-medium Phase Shift and Cross Section ................ 156 6.4 In-medium Low-Energy Parameters ................... 161 6.5 Medium Effect vs. Parameter Variation ................. 166 6.6 Conclusion ................................. 167 7 Summary and outlook 171 Appendices 175 A Coupling Constant Values 177 CONTENTS 11 B Partial Wave Potential Matrix Elements 179 C LSJM Representation Operators 183 D Partial Wave Decomposition of NN 187 E Helicity State Basis Decomposition 189 Bibliography 192 Preface \Not only is the Universe stranger than we think, it is stranger than we can think. " Werner Heisenberg About a century ago, microscopic physics started with the demand of under- standing atomic spectra. That effort gave rise to the invention of quantum mech- anics, the key science of our time. From the present day's point of view, theoretical as well as experimental studies of atoms and molecules have become standard work, reaching even deep into the industrial and commercial sectors. That is because those systems exist under the action of the meanwhile well known electromagnetic laws of force. In nuclear and hadron physics, however, we have not yet reached a comparable state of knowledge. In addition to the complexities of quantum mechanics, nuclear physics is governed by the strong force which in its low energy realization is a highly complicated non-perturbative phenomenon. While the nuclear sector has taken a large step forward to produce a good number of successful, realistic, high- precision models [10, 11, 55, 62, 64, 110] utilizing a rich data set available with simultaneous