Electromagnetic Properties of Light and Heavy Baryons in the Relativistic Quark Model Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften der Fakult¨atf¨urMathematik und Physik der Eberhard-Karls-Universit¨atzu T¨ubingen vorgelegt von Diana Nicmorus Marinescu aus Ploiesti (Rum¨anien) 2007 Tag der m¨undlichen Pr¨ufung:14. Juni 2007 Dekan: Prof. Dr. Nils Schopohl 1. Berichterstatter: Prof. Dr. Thomas Gutsche 2. Berichterstatter: Prof. Dr. Dr. h.c. mult. Amand F¨aßler Suena de capao, corazon cabao .... Abstract One of the main challenges of nowadays low-energy physics remains the description of the internal structure of hadrons, strongly connected to the electromagnetic properties of matter. In this vein, the success of the relativistic quark model in the analysis of the hadron structure constitutes a solid motivation for the study carried out throughout this work. The relativistic quark model is extended to the inves- tigation of static electromagnetic properties of both heavy and light baryons. The bare contributions to the magnetic moments of the single-, double- and triple-heavy baryons are calculated. Moreover, the relativistic quark model allows the study of the electromagnetic properties of the light baryon octet incorporating meson cloud contributions in a perturbative manner. The long disputed values of the multipole ratios E2/M1 and C2/M1 and the electro- magnetic form factors of the N ∆γ transition are successfully reproduced. The relativistic quark model→ can be viewed as a quantum field theory approach based on a phenomenological Lagrangian coupling light and heavy baryons to their constituent quarks. In our approach the baryon is a compos- ite object of three constituent quarks, at least in leading order. The effective interaction Lagrangian is written in terms of baryon and constituent quark fields. The effective action preserves Lorentz covariance and gauge invari- ance. The main ingredients of the model are already introduced at the level of the interaction Lagrangian: the three-quark baryon currents, the Gaussian distribution of the constituent quarks inside the baryon and the composite- ness condition which sets an upper limit for the baryon-quark vertex. The S-matrix elements are expressed by a set of Feynman quark-diagrams. The model contains only few parameters, namely, the cut-off parameter of the Gaussian quark distribution and the free quark propagator, which are un- ambiguously determined from the best fit to the data. The heavy quark limit within this model reveals an exact agreement in leading order with the model-independent predictions for the magnetic moments of the heavy baryons. For the light sector, a Lorentz covariant chiral quark Lagrangian is used to dress the constituent quarks by pseudoscalar meson clouds. The main achievement consists in the factorization of the valence quark contribu- tions and the meson cloud contributions in the calculation of electromagnetic properties of light baryons. Abstrakt Eine der Hauptherausforderungen der heutigen Niederenergiephysik besteht in der Beschreibung der internen Struktur von Hadronen, welche im en- gen Zusammenhang mit den elektromagnetischen Eigenschaften der Ma- terie zusammenh¨angt.Der vorherige Erfolg des relativistischen Quarkmod- ells in der Analyse der Hadronenstruktur stellt eine solide Grundlage f¨ur die Untersuchungen, welche in dieser Arbeit durchgef¨uhrtwerden, dar. In diesem Zugang wird das Baryon, zumindest in f¨uhrenderOrdnung, durch den Bindungszustand aus Konstituentenquarks beschrieben. Das relativistische Quarkmodell wird in der vorliegenden Arbeit erweitert, um die statischen, elektromagnetischen Eigenschaften von schweren und le- ichten Baryonen zu untersuchen. Das relativistische Quarkmodell basiert auf einen quantenfeldtheoretischen Zugang, welcher, in einer ph¨anomenologischen Lagrange-Dichte formuliert, die Kopplung von leichten und schweren Bary- onen an die Konstituentenquarks beschreibt. Die entsprechende effektive Wirkung erh¨altLorentz-Kovarianz und Eichinvarianz. Die Hauptannahmen des Modells werden auf dem Niveau der Wechselwirkungs Lagrange-Dichte eingef¨uhrt:Drei-Quark Baryonen-Str¨ome,die Gauss-Verteilung der Konsti- tutentenquarks im Baryon und die sogenannte ’compositeness condition’, welche die Kopplung der nackten Baryonen an die Konstituentenquarks bes- timmt. Die entsprechenden S-Matrix Elemente werden durch einen Satz von Feynman-Quark-Diagrammen ausgedr¨uckt. Das relativistische Quarkmodell beinhaltet einen Satz von wenigen Parametern: Breite der Gaussverteilung und Konstituentenquarkmassen, welche durch Anpassung an vorherige Daten fixiert werden. Die Valenzquarkbeitr¨agezu den magnetischen Mometen der einfach, dop- pelt und dreifach schweren Baryonen werden berechnet. Im Limes schw- erer Quarkmassen kann innerhalb dieses Modells in f¨uhrenderOrdnung ex- akte Ubereinstimmung¨ mit den modell-unabh¨angigenVorhersagen zu den magnetischen Momenten schwerer Baryonen erreicht werden. Im leichten Baryonen-Sektor wird zus¨atzlich ein Lorentz kovariante, chirale Quark-La- grange-Dichte genutzt, um die See-Quark Beitr¨age,ausgedr¨uckt durch pseu- doskalare Mesonen, in Konsistenz mit chiraler Symmetrie einzubinden. Die chirale Lagrange-Dichte wird zur Ordnung O(p4) und in der Einschleifen- N¨aherungausgewertet, um die durch die chiralen Effekte angezogenen Quark- operatoren zu bestimmen. Das Hauptergebnis dieser Technik ist, dass die Beitr¨ageder Mesonen und der Valenzquarks in den elektromagnetischen, baryonischen Amplituden faktorisieren. Anwendungen dieser chiralen Er- weiterung des relativistischen Quarkmodell auf elektromagnetischen Eigen- schaften des leichten Baryonen-Oktetts werden ausgearbeitet. Zus¨atzlich studieren wir die elektromagnetischen N ∆ Uberg¨ange,wo¨ relativistische Effekte eine entscheidende Rolle spielen.− Insbesondere k¨onnendie Multi- polverh¨altnisse E2/M1, C2/M1 und die elektromagnetischen Formfaktoren des radiativen Ubergangs¨ N ∆γ erfolgreich reproduziert werden. → Contents 1 Introduction 9 1.1 Models of the baryons . 10 1.2 Chiral Perturbation Theory . 14 1.3 Heavy Quark Effective Theory . 23 2 Relativistic Three Quark Model 29 2.1 General overview . 29 2.2 Three-quark baryon currents . 33 2.3 Compositeness condition . 45 2.4 Electromagnetic interaction . 47 3 Magnetic moments of heavy baryons 51 3.1 Baryon mass operator and matrix elements . 58 3.2 Heavy-quark limit and matching to Heavy Hadron ChPT . 63 3.3 Numerical results and discussion . 66 4 Magnetic moments of light baryons 75 4.1 Matrix elements of valence quark operators . 83 4.2 Matching to ChPT . 84 4.3 Light baryon magnetic moments . 86 5 N ∆γ transition 101 − 6 Summary 121 A Representations of the Lorentz group 127 7 8 CONTENTS B Three-quark currents 131 B.1 Light baryons . 131 B.2 Heavy baryons . 132 B.3 Baryon wave functions . 134 C Feynman rules for the non-local electromagnetic vertex 137 D Calculation of matrix elements 141 E Non-relativistic spin-flavor wave functions and magnetic mo- ments 145 F Relativistic form factors for the N ∆γ transition 147 → G Uniqueness of the ∆+(1232) three-quark current 149 Chapter 1 Introduction In this work we investigate the structure of baryons applying a relativistic constituent quark model. Properties of both light and heavy baryons are studied in a quantum field theory approach [1, 2]. Baryons are systems governed by the strong interaction. We aim to de- scribe such interactions introducing a phenomenological Lorentz covariant and gauge invariant Lagrangian. Due to the success of the naive quark model predictions for static baryon properties [3, 4, 5, 6] and motivated by the re- sults of deep inelastic lepton scattering, it is reasonable to think of baryons in leading order as composite systems of three quarks. The prototype of a such system is the nucleon. The model we describe here has been already developed and applied to the nucleon, and later on improved for a series of low energy processes obeyed by other heavier states [7, 8, 9, 10, 11, 12]. Among various attempts to describe the low-energy behaviour of nucleons in a relativistic framework, our approach has particular advantages: it is rather intuitive, employs a solid and within quantum field theory consistent calculational technique, has a fixed and small number of parameters. Our results are in good agreement with other models and effective approaches. Moreover, our approach recovers model independent predictions, as the ones offered by chiral perturbation theory and heavy quark effective theory. Having in mind the importance and success of these approaches we will start by giving a short overview on the naive quark models, on the basics of chiral perturbation theory and heavy quark symmetry. We continue with an overview of the main issues of the relativistic quark model, followed by two applications: the calculation of magnetic moments of single-, double-, triple- heavy baryons and the calculation of magnetic moments of light baryons, 9 10 CHAPTER 1. INTRODUCTION taking into consideration meson cloud effects. A challenging aspect is the study of the N ∆ transition, which we will present in the last part of this work. → The nucleon and its observed detailed properties are of great impor- tance to understand and test the theory of strong interaction. A theoretical study of the nucleon collects phenomena associated with QCD (confinement, spontaneous symmetry breaking, and asymptotic freedom) as well as phe- nomenological approaches, that is models. At low energies it is possible to obtain model independent predictions