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In-Medium QCD Sum Rules for Ω Meson, Nucleon and D Meson QCD-Summenregeln F ¨Ur Im Medium Modifizierte Ω-Mesonen, Nukleonen Und D-Mesonen

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In-Medium QCD Sum Rules for Ω Meson, Nucleon and D Meson QCD-Summenregeln F ¨Ur Im Medium Modifizierte Ω-Mesonen, Nukleonen Und D-Mesonen Institut f¨ur Theoretische Physik Fakult¨at Mathematik und Naturwissenschaften Technische Universit¨at Dresden In-Medium QCD Sum Rules for ω Meson, Nucleon and D Meson QCD-Summenregeln f ¨ur im Medium modifizierte ω-Mesonen, Nukleonen und D-Mesonen Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium vorgelegt von Ronny Thomas geboren am 7. Februar 1978 in Dresden C Dresden 2008 To Andrea and Maximilian Eingereicht am 07. Oktober 2008 1. Gutachter: Prof. Dr. Burkhard K¨ampfer 2. Gutachter: Prof. Dr. Christian Fuchs 3. Gutachter: PD Dr. Stefan Leupold Verteidigt am 28. Januar 2009 Abstract The modifications of hadronic properties caused by an ambient nuclear medium are investigated within the scope of QCD sum rules. This is exemplified for the cases of the ω meson, the nucleon and the D meson. By virtue of the sum rules, integrated spectral densities of these hadrons are linked to properties of the QCD ground state, quantified in condensates. For the cases of the ω meson and the nucleon it is discussed how the sum rules allow a restriction of the parameter range of poorly known four-quark condensates by a comparison of experimental and theoretical knowledge. The catalog of independent four-quark condensates is covered and relations among these condensates are revealed. The behavior of four-quark condensates under the chiral symmetry group and the relation to order parameters of spontaneous chiral symmetry breaking are outlined. In this respect, also the QCD condensates appearing in differences of sum rules of chiral partners are investigated. Finally, the effects of an ambient nuclear medium on the D meson are discussed and relevant condensates are identified. Kurzfassung Die Ver¨anderungen von Hadroneneigenschaften durch ein umgebendes nukleares Medium (Kern- materie) werden mit der Methode der QCD-Summenregeln untersucht. Dies wird am Beispiel des ω-Mesons, des Nukleons und des D-Mesons vorgef¨uhrt. Durch die Summenregeln werden inte- grierte Spektraldichten dieser Hadronen in Beziehung zu Eigenschaften des QCD-Grundzustandes, quantifiziert in Kondensaten, gesetzt. Diskutiert wird am Beispiel des ω-Mesons und des Nuk- leons, wie diese Summenregeln eine Einschr¨ankung des Parameterbereiches von wenig bekannten Vierquark-Kondensaten durch Vergleich von experimentellen und theoretischen Erkenntnissen er- lauben. Ein Katalog unabh¨angiger Vierquark-Kondensate wird aufgestellt und Relationen zwi- schen diesen Kondensaten werden deutlich gemacht. Das Verhalten der Vierquark-Kondensate unter der chiralen Symmetriegruppe und der Zusammenhang mit Ordnungsparametern spontaner chiraler Symmetriebrechung werden behandelt. In dieser Hinsicht werden auch die in Diffe- renzen der Summenregeln chiraler Partner eingehenden QCD-Kondensate untersucht. Schließlich werden die Effekte endlicher Kerndichten beim D-Meson diskutiert und relevante Kondensate identifiziert. Contents 1 HadronsinMedium................................... ......................... 7 1.1 Probing Strongly Interacting Matter . ........ 7 1.2 Experimental Status and Perspectives . ........ 9 2 QCDSumRulesinMedium............................... ..................... 13 2.1 Motivation: The Vacuum Case . 13 2.2 Generalization to Medium . 17 2.2.1 Dispersion Relations . 17 2.2.2 Weighted Spectral Moments . 20 2.2.3 Operator Product Expansion . 22 2.3 QCDCondensates.................................. 24 2.3.1 SymmetriesofQCD ............................ 24 2.3.2 Catalog of QCD Condensates . 30 2.4 Four-Quark Condensates . 34 2.4.1 Projection and Classification . 35 2.4.2 Factorization and Parametrization of Four-Quark Condensates . 37 2.4.3 Four-Quark Condensates as Chiral Order Parameters . ......... 42 3 AnalysisofQCDSumRules............................. ...................... 51 3.1 Light Vector Mesons: ω Meson........................... 51 3.1.1 HadronicModels .............................. 51 3.1.2 QCDSumRule............................... 53 3.1.3 Constraints on Four-Quark Condensates . ...... 55 3.2 Light-Quark Baryons: Nucleon . ..... 61 3.2.1 QCDSumRuleEquations ..... ..... ...... ..... .... 62 3.2.2 Impact of Four-Quark Condensates . 66 3.3 Pseudoscalar Heavy-Light Quark Mesons: D Meson ............... 77 3.3.1 Operator Product Expansion for Heavy-Light Quark Systems....... 77 3.3.2 QCD Sum Rules for the D Meson..................... 79 3.3.3 Mass splitting of D+ and D− ....................... 80 4 Summary........................................... ............................ 83 Appendix........................................... ............................ 85 A BorelTransforms................................... ........................... 85 B Operator Product Expansion Techniques................ ..................... 87 6 Contents C Addendum: Four-Quark Condensates.................... ..................... 91 C.1 Alternative Derivation of Pure-Flavor Four-Quark Condensate Interrelations . 91 C.2 Four-Quark Expectation Values in the Nucleon . ......... 92 C.3 BasisTransformations . 92 D ListofAcronyms.................................... ........................... 95 Bibliography....................................... ............................ 99 1 Hadrons in Medium 1.1 Probing Strongly Interacting Matter Hadrons are the observed physical degrees of freedom of the strong interaction at low temperatures and densities or in vacuum [1]. However, it is necessary to relate the measured properties of hadrons to the fundamental theory of the strong interaction - Quantum Chromodynamics (QCD). The degrees of freedom in this local gauge theory are the quarks and, as gauge bosons, the gluons. These fields are said to be confined in hadrons, since quarks and gluons have not been observed as asymptotically isolated particles (confinement hypothesis). Nevertheless, the mass of a hadron, formed out of quarks and gluons, should be explained by QCD parameters. Since the sum of the current quark masses is orders of magnitude below the hadron masses, the latter must be generated from the interaction energy in the theory of QCD. The hadron masses are thus related to the properties of the QCD ground state, which has a highly non-trivial structure. A relation between QCD parameters and the hadronic spectrum is impeded by the generic problems of low-energy QCD. In the high energy limit, many observables can be calculated within perturbation theory where the running coupling is small enough to allow for a perturbative expansion of scattering probabilities due to asymptotic freedom. While going subsequently to lower energy scales, the running coupling αs increases and the expansion in powers of the coupling strength becomes invalid. Suitable tools to explore the structure of low-energy QCD in relation to properties of hadrons are among others ”lattice QCD”, chiral effective field theory, instanton models and QCD sum rules. The former is based on a computational treatment of the theory on a discretized space-time lattice with sophisticated algorithms. With improving computational resources it approaches a realistic prediction of QCD observables. Unfortunately, the numerical results obtained therefrom often have to be extrapolated to physical values of the quark masses. The second approach con- tains effective fields for the hadrons and suitable interaction terms from which self-energies and scattering processes etc. can be evaluated once a unique fit of unknown constants is at our dis- posal. Instanton models describe special features of the QCD ground state by the density and size of instanton configurations, which are classical solutions of the equations of motion. In this work, the approach of QCD sum rules (QSR) is focused on. It was originally formulated by Shifman, Vainshtein and Zakharov [2, 3] to describe for example masses of light vector mesons in vacuum [4]. The method since then gained attention in numerous applications, e.g. to calculate masses and couplings of low-lying hadrons, magnetic moments, etc. (cf. [5–7]). Its particular meaning is that numerous hadronic observables are directly linked to a set of fundamental QCD quantities, the condensates. Condensates are ground state expectation values of QCD and so they quantify the complexity of the QCD ground state. In QCD sum rules, the condensate terms occur as power corrections of a perturbative expansion and dominate the low-energy behavior of QCD. One expects then that condensates should determine the properties of hadrons as well. Although the method does not claim to be a precision tool it is quite meaningful since it can be applied to numerous hadronic observables using a unique set of condensates. Furthermore some of these condensates are directly linked to symmetries of QCD and can measure symmetry violations. 8 1 Hadrons in Medium Especially useful becomes this approach when one is interested in changes of hadron properties. Such modifications can arise in a medium of non-vanishing temperature or density. The investi- gation of these modifications sheds light on the complex ground state of QCD. Changes in this ground state are expected to be reflected in its excitation spectrum, analogous to the changes of atomic excitation spectra embedded in external electromagnetic fields (Stark and Zeeman effects). The excitations of the QCD ground state are the hadrons, and the corresponding external field is represented by the surrounding nuclear matter. Hadrons in the nuclear medium therefore probe this ground state by the modifications they perceive (see Fig. 1.1). Figure 1.1: Illustration of the modification of hadronic
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