Microscopic baryon-baryon interactions at finite density and hypernuclear structure [Elektronische Ressource] / vorgelegt von Christoph Marcus Keil

Sujets

• Physik

Informations

• Publié par: justus-liebig-universitat_giessen
• Publié le: 01 janvier 2006
• Nombre de lectures: 6
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

Microscopic baryon-baryon interactions
at finite density and hypernuclear
structure
Dissertation
zur
Erlangung des Doktorgrades
der Naturwissenschaftlichen Fakult¨at
der Justus-Liebig-Universit¨at Gießen
Fachbereich 7 – Mathematik, Physik, Geographie
vorgelegt von
Christoph Marcus Keil
aus Linden
Gießen, 2004Dekan: Prof. Dr. Volker Metag
I. Gutachter: Prof. Dr. Horst Lenske
II. Gutachter: Prof. Dr. Werner Scheid
Tag der mu¨ndlichen Pru¨fung: 20.12.2004Contents
Introduction 1
I. Relativistic ab-initio Calculations 9
1. Relativistic Scattering Theory 11
1.1. Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2. Symmetries and systematics of the T-matrix . . . . . . . . . . . . . . . . 13
1.2.1. Partial wave decomposition . . . . . . . . . . . . . . . . . . . . . 13
1.2.2. The structure of the T-matrix . . . . . . . . . . . . . . . . . . . . 14
1.2.3. Scattering of identical particles . . . . . . . . . . . . . . . . . . . 17
1.3. 3D-reduced two baryon propagators . . . . . . . . . . . . . . . . . . . . . 20
1.3.1. Reference frames in two particle scattering . . . . . . . . . . . . . 20
1.3.2. The pseudo-potential equation . . . . . . . . . . . . . . . . . . . . 23
1.3.3. The Blankenbecler-Sugar propagator . . . . . . . . . . . . . . . . 23
1.3.4. The Thompson propagator . . . . . . . . . . . . . . . . . . . . . . 27
1.3.5. Discussion of 3D propagators . . . . . . . . . . . . . . . . . . . . 27
1.4. The K-matrix approximation and scattering phase shifts . . . . . . . . . 28
1.4.1. Scattering Phase shifts in multi-channel systems . . . . . . . . . . 29
2. Relativistic Meson-Exchange Models 31
2.1. Invariant Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2. Calculation of effective interactions . . . . . . . . . . . . . . . . . . . . . 35
2.2.1. Regularization of the loop integrals . . . . . . . . . . . . . . . . . 37
2.2.2. Multi baryon coupled channel calculations . . . . . . . . . . . . . 37
3. Microscopic In-Medium Interaction 41
3.1. In-medium scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1. The Pauli operator . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2. The relativistic structure of the T-matrix . . . . . . . . . . . . . . 51
3.1.3. Self-energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2. Relativistic mean-field kinematics . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1. Reference frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3. Relativistic mean-field dynamics – saturation . . . . . . . . . . . . . . . . 61
iContents
4. The Density Dependent Relativistic Hadron Field Theory 67
4.1. The DDRH formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2. Microscopic vertices in DDRH . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1. The structure of the Λ-meson vertex . . . . . . . . . . . . . . . . 71
4.3. Mean-field dynamics in Λ hypernuclei . . . . . . . . . . . . . . . . . . . . 73
4.3.1. The Λ-ω tensor interaction . . . . . . . . . . . . . . . . . . . . . . 73
5. The Dynamics of Effective ΛN Interactions 75
5.1. ΛN interactions in free space . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2. ΛN interactions at finite density . . . . . . . . . . . . . . . . . . . . . . . 78
5.3. Consequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6. The Vertex Renormalization Approach 85
6.1. Formal developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2. A schematic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2.1. Free space scattering . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.2. Interactions at finite density . . . . . . . . . . . . . . . . . . . . . 89
6.3. Discussion of the vertex renormalization . . . . . . . . . . . . . . . . . . 93
II. Hypernuclear Structure 95
7. Hypernuclear Physics 97
7.1. Hypernuclear experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2. Hypernuclear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8. Spectra of Hypernuclei with High-Spin Core States 103
8.1. The conventional data analysis. . . . . . . . . . . . . . . . . . . . . . . . 103
8.2. Hyperon-nucleon coupling constants in medium-mass nuclei . . . . . . . . 104
89 518.3. Reexamination of Y and V data . . . . . . . . . . . . . . . . . . . . . 106Λ Λ
8.4. Determination of the Λ vertices in DDRH theory . . . . . . . . . . . . . 111
8.5. Consequencies and recommendation . . . . . . . . . . . . . . . . . . . . . 113
9. The Hypernuclear Auger Effect 117
9.1. Modeling the Hypernuclear Auger Effect . . . . . . . . . . . . . . . . . . 118
9.2. Results for the hypernuclear Auger effect . . . . . . . . . . . . . . . . . . 120
2099.2.1. Pb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Λ
919.2.2. Zr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Λ
9.3. Resum´e on Auger spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 131
10.Summary and Outlook 133
A. Definitions and Conventions 139
A.1. Space-time metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2. The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2.1. Dirac matrices and traces . . . . . . . . . . . . . . . . . . . . . . 139
iiContents
A.3. Lorentz boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Appendix 139
B. Meson Exchange Models 141
B.1. Helicity matrix elements of Born diagrams . . . . . . . . . . . . . . . . . 141
B.1.1. Definitions and conventions . . . . . . . . . . . . . . . . . . . . . 141
B.1.2. Helicity matrix elements . . . . . . . . . . . . . . . . . . . . . . . 144
B.2. Partial wave decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.2.1. Properties of d functions . . . . . . . . . . . . . . . . . . . . . . . 147
B.2.2. Partial wave decomposition of helicity matrix elements . . . . . . 148
B.3. The Bonn potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C. G-Matrix: Details 151
C.1. Decomposition of the G-matrix . . . . . . . . . . . . . . . . . . . . . . . 151
C.1.1. Removal of kinematic singularities in the T-matrix decomposition 151
C.1.2. Matrix elements of covariants . . . . . . . . . . . . . . . . . . . . 153
D. DDRH Parameter Sets 155
D.1. Nucleon-nucleon interactions . . . . . . . . . . . . . . . . . . . . . . . . . 155
D.2. Hyperon-nucleon interactions . . . . . . . . . . . . . . . . . . . . . . . . 155
E. Hypernuclear Structure 157
E.1. Matrix Elements for Auger neutron rates . . . . . . . . . . . . . . . . . . 157
F. Numerics 159
F.1. Solution of the Bethe-Salpeter integral equation . . . . . . . . . . . . . . 159
F.1.1. Numerical evaluation of principle value integrals . . . . . . . . . . 161
Bibliography 163
Deutsche Zusammenfassung 171
iiiContents
ivIntroduction
The interaction between baryons, of which protons and the neutrons are the lightest
and best known, is very strong. This does not only provide a variety of very interesting
phenomena, but requires also an elaborate framework to describe it. The interaction
between baryons in a baryonic medium is a special challenge, it changes dramatically,
depending on the density and composition of the medium. From a modern point of
view these interactions observed at finite density or between baryons in free space are
only effective interactions, different facets of a more fundamental interaction between
the particles, from which the effective interactions can be derived in one consistent for-
malism. The underlying bare or microscopic interaction, gouverned by quantum chro-
modynamics (QCD), cannot be accessed directly, but has to be traced back using its
various appearences. In this work we are going to develop a microscopic model, describ-
ing baryon-baryon interactions in free space, in infinite, homogeneous systems of finite
density and in small, nuclear systems.
The interaction between baryons is not only very strong, but also of very short range,
−15about a few of 10 m. It is, however, in large parts responsible for the structure of the
matter surrounding us – at all scales from close by, in our environment to far away, in
the whole visible universe. Baryon-baryon interactions connect very largeand very small
scales. To get a taste of where these are at work all around us and to see their relevance
in our world, let us start with a short look into the history of baryons in the universe
and point out the places in which their interactions are of importance.
Baryons – the constituents of the matter surrounding us
Baryons are as old as the universe itself, they were created already 100 seconds after the
big-bang, when the hot soup of quarks and gluons, from which baryons are made, cooled
down so far that they started sticking together in tiny lumps of quarks [Kolb90]. Due
to the confining character of the quark-quark interaction only baryons, bags containing
three valence quarks, were left. And maybe also heavier quark bags, the strangelets,
which, however, would interact very weakly and have so far not been observed. Due
to processes violating CP symmetry, which regulates the balance between matter and
antimatter, a tiny amount ofbar