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Baryons As Relativistic Bound States of Quark and Diquark

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Baryons As Relativistic Bound States of Quark and Diquark Baryons as Relativistic Bound States of Quark and Diquark DISSERTATION zur Erlangung des Grades eines Doktors der Naturwissenschaften der Fakultat¨ fur¨ Physik der Eberhard-Karls-Universitat¨ zu Tubingen¨ vorgelegt von MARTIN OETTEL aus Freiberg i. S. 2000 Tag der mundlichen¨ Prufung:¨ 20. Oktober 2000 Dekan: Prof. Dr. G. Wagner 1. Berichterstatter: PD Dr. R. Alkofer 2. Berichterstatter: Prof. Dr. H. Muther¨ Zusammenfassung In der vorliegenden Dissertation wird ein Modellrahmen zur Beschreibung von Baryonen als Diquark-Quark-Bindungszustande¨ vorgestellt, der auf eine mani- feste Kovarianz der Beschreibung zielt. Nach einem kurzen Uberblick¨ uber¨ drei oft verwandte Typen von Baryonmo- dellen wird das relativistische Problem eines Drei-Quark-Bindungszustandes unter Vernachlassigung¨ von dreiteilchenirreduziblen Wechselwirkungen betrachtet. Di- quarks werden als separable Korrelationen in der Zwei-Quark-Korrelationsfunk- tion eingefuhrt.¨ Mit diesen hier angenommenen Vereinfachungen gelangt man zu einer Beschreibung der Baryonen mittels einer Bethe-Salpeter-Gleichung fur¨ Bindungszustande¨ von Quark und Diquark, wobei die beiden Konstituenten uber¨ Quarkaustausch miteinander wechselwirken. Skalare und axialvektorielle Di- quarks werden in Analogie zum Massenspektrum der Mesonen als die dominan- ten Zwei-Quark-Korrelationen in den Baryonen angenommen. Die Struktur der relativistischen Wellenfunktion, die die skalaren und axialvektoriellen Korrelatio- nen beschreibt, wird unter Zuhilfenahme einer Partialwellenzerlegung ausfuhrlich¨ diskutiert. Die numerischen Losungen¨ der kovarianten Bethe-Salpeter-Gleichung wer- den des weiteren zur Berechnung von elektromagnetischen, starken und axialen Nukleon-Formfaktoren herangezogen. Hierbei wird der elektromagnetische Strom- operator in dem Modell so konstruiert, daß er die Eichinvarianz respektiert. In- varianz unter chiralen Symmetrietransformationen ist dagegen leicht verletzt, da Vektordiquarks vernachlassigt¨ wurden. Dies fuhrt¨ zu einer moderaten Verletzung der Goldberger-Treiman-Beziehung. Weiterhin wird die Moglichk¨ eit, Confinement effektiv uber¨ geeignete Modi- fikationen der Quark- und Diquarkpropagatoren zu parametrisieren, untersucht. Mittels einer solchen Modifikation kann das Massenspektrum der Oktett- und Dekuplettbaryonen berechnet werden. Obwohl diese Vorgehensweise der Con- finement-Parametrisierung sich als geeignet fur¨ die Berechnung raumartiger Nuk- leoneigenschaften erweist, versagt sie bei der Behandlung von Prozessen, bei denen große Energien auf das Nukleon ubertragen¨ werden, wie z.B. bei aus- gewahlten¨ Mesonproduktionsprozessen. In einem weiteren Abschnitt werden die Losungen¨ fur¨ Nukleonwellenfunktio- nen und -observablen miteinander verglichen, die in der voll relativistischen Be- handlung und in einer weitverbreiteten semirelativistischen Naherung¨ gewonnen werden. Die hierbei beobachteten betrachtlichen¨ Abweichungen fur¨ die berech- neten Observablen, die sich aus der Anwendung der beiden Methoden ergeben, zeigen die Unzulanglichk¨ eiten der semirelativistischen Behandlung auf. Abstract In this thesis a model framework for describing baryons as diquark-quark bound states is presented which is formulated in an explicitly covariant manner. After a short summary of three widely used classes of baryon models the rel- ativistic bound state problem for three quarks is considered, where three-particle irreducible interactions are neglected. Diquarks are introduced as separable cor- relations in the two-quark correlation function. With these simplifications as- sumed, baryons are described by a Bethe-Salpeter equation for bound states of quark and diquark which interact by quark exchange. In analogy to the meson spectrum, scalar and axialvector diquarks are considered to be the most important two-quark configurations within baryons. The structure of the relativistic wave function which encompasses the scalar and axialvector correlations is discussed in terms of a partial wave decomposition. The numerical solutions of the covariant Bethe-Salpeter equation are subse- quently employed in the calculation of electromagnetic, strong and axial form factors of the nucleons. The construction of the electromagnetic current opera- tor in the diquark-quark model respects gauge invariance. Invariance under chiral symmetry transformations is slightly violated for vector diquarks have been ne- glected, leading to a modest violation of the Goldberger-Treiman relation. The possibility of an effective parametrization of confinement by suitable modifications of the quark and diquark propagators is investigated. Thereby the mass spectrum of octet and decuplet baryons can be calculated. It is shown that the modelling of confinement chosen here is suitable for the calculation of space- like nucleon properties, but its applicability breaks down for processes where large energies are transferred to the nucleon as it is the case in certain meson production processes. Model solutions for nucleon wave functions and observables are compared between the full relativistic treatment and a widely used semi-relativistic approx- imation. The considerable deviations between the calculated observables using the two methods illustrate the inadequacy of employing the semi-relativistic treat- ment. Contents 1 Introduction 7 1.1 General remarks . 7 1.2 Modelling Baryons . 8 1.2.1 Constituent quark models . 8 1.2.2 Soliton models . 10 1.2.3 Bag models . 14 1.2.4 The diquark-quark picture: baryons as relativistic bound states . 16 2 The Covariant Diquark-Quark Model 19 2.1 Reduction of the relativistic three-quark problem . 19 2.1.1 Distinguishable quarks . 20 2.1.2 Identical quarks . 25 2.2 Diquark correlations . 28 2.3 The diquark-quark Bethe-Salpeter equation . 34 2.3.1 Nucleon . 35 2.3.2 Delta . 36 2.4 Decomposition of the Faddeev amplitudes . 37 2.4.1 Partial wave decomposition . 43 2.5 Numerical solutions . 49 2.5.1 Numerical method . 49 2.5.2 Solutions I: The scalar diquark sector . 54 2.5.3 Solutions II: Scalar and axialvector diquarks . 60 3 Electromagnetic Form Factors 65 3.1 The nucleon current operator in the diquark-quark model . 67 3.1.1 Impulse approximation . 71 3.1.2 Seagulls . 74 3.1.3 Normalization of the Bethe-Salpeter wave function and the nucleon charges . 78 3.1.4 Summary . 81 6 Contents 3.2 Numerical calculations . 82 3.2.1 Numerical method . 82 3.2.2 Results . 84 3.2.3 Conclusions . 89 4 Strong and Axial Form Factors 91 4.1 Scalar and vector/axialvector diquarks as chiral multiplets . 94 4.2 Pseudoscalar and pseudovector current operator . 95 4.3 Numerical results . 99 5 Effective Confinement 103 5.1 Confinement in model propagators . 104 5.2 The octet-decuplet mass spectrum . 106 5.3 Electromagnetic form factors . 109 5.4 Discussion . 113 6 The Salpeter Approximation 117 6.1 Solutions for vertex functions . 118 6.2 Results for observables . 121 7 Summary and Conclusions 127 A Varia 131 A.1 Conventions . 131 A.2 Color and flavor factors in the Bethe-Salpeter equations . 133 A.2.1 Nucleon . 133 A.2.2 . 135 A.3 Wave functions . 135 A.4 Bethe-Salpeter equations for octet and decuplet . 136 B Form Factor Calculations 141 B.1 Resolving diquarks . 141 B.1.1 Electromagnetic Vertices . 141 B.1.2 Pseudoscalar and pseudovector vertices . 143 B.2 Calculation of the impulse approximation diagrams . 145 Bibliography 151 Chapter 1 Introduction 1.1 General remarks Up to now, no convincing solution for the problem of describing baryons and mesons in terms of a few underlying fields has been found, yet it is widely be- lieved that quarks and gluons are the basic entities which build the hadrons. The associated theory is called Quantum Chromodynamics (QCD). The complexity of this theory in the low energy sector is prohibitive for a straightforward deduc- tion of hadronic properties from it. Therefore, models with a simplified dynamics have been employed over the last decades which aim at describing hadrons with degrees of freedom borrowed from QCD (like quarks) or from the observable par- ticle spectrum (like pions) that might be better suited in the treatment of hadrons. The aim of the present thesis is to study a relativistic framework for the de- scription of baryons employing quark and diquark degrees of freedom. Actual calculations are performed with rather simple assumptions about these, but it is hoped that further progress in knowledge about QCD’s quark propagator and the structure of 2-quark correlations will clarify whether the reduction to quarks and diquarks in the treatment of baryons is compatible with QCD. We refrain from discussing QCD here. The relevance of QCD and its (approx- imate) symmetries for hadronic physics is discussed in many good textbooks on quantum field theory, see e.g. refs. [1, 2]. In the remaining part of this Introduction we will give a short description of the basic ideas underlying three classes of hadronic models which have become popular (and have remained so) over the last years. This brief survey is by no means meant to be exhausting, and serves only as a motivation to consider a fully covariant approach like the diquark-quark model. A short overview about the topics which are covered in the subsequent chapters is given in the last subsection of this chapter. 8 Introduction 1.2 Modelling Baryons 1.2.1 Constituent quark models In the early sixties, it has become recognized that the eight spin-1/2 baryons today referred to as “octet baryons” belong to an eight-dimensional tensor representation of the unitary group ¡£¢¥¤§¦©¨ . Obviously this unitary symmetry is
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