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Medium Modifications of Mesons

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Medium Modifications of Mesons Medium Modifications of Mesons Chiral Symmetry Restoration, in-medium QCD Sum Rules for D and ρ Mesons, and Bethe-Salpeter Equations Dissertation zur Erlangung des akademischen Grades Doctor rerum naturalium vorgelegt von Thomas Uwe Hilger geboren am 29.09.1981 in Eisenhüttenstadt Institut für Theoretische Physik Fachrichtung Physik Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden Dresden 2012 Eingereicht am 11.04.2012 1. Gutachter: Prof. Dr. B. Kämpfer 2. Gutachter: Prof. Dr. S. Leupold Kurzdarstellung Das Zusammenspiel von Hadronen und Modifikationen ihrer Eigenschaften auf der einen Seite und spontaner chiraler Symmetriebrechung und Restauration auf der anderen Seite wird untersucht. Es werden die QCD Summenregeln für D und B Mesonen in kalter Materie berechnet. Wir bestimmen die Massenaufspaltung von D D¯ und B B¯ Mesonen als Funktion der Kerndichte und untersuchen den − − Einfluss verschiedener Kondensate in der Näherung linearer Dichteabhängigkeit. Die Analyse beinhaltet ∗ ebenfalls Ds und D0 Mesonen. Es werden QCD Summenregeln für chirale Partner mit offenem Charm- freiheitsgrad bei nichtverschwindenden Nettobaryonendichten und Temperaturen vorgestellt. Es wird die Differenz sowohl von pseudoskalaren und skalaren Mesonen, als auch von Axialvektor- und Vek- tormesonen betrachtet und die entsprechenden Weinberg-Summenregeln hergeleitet. Basierend auf QCD Summenregeln werden die Auswirkungen eines Szenarios auf das ρ Meson untersucht, in dem alle chiral ungeraden Kondensate verschwinden wohingegen die chiral symmetrischen Kondensate ihren Vaku- umwert behalten. Die komplementären Folgerungen einer Massenverschiebung und Verbreiterung der ρ Mesonanregung werden diskutiert. Ein alternativer Zugang basierend auf gekoppelten Dyson-Schwinger- und Bethe-Salpeter-Gleichungen für Quarkbindungszustände wird untersucht. Zu diesem Zwecke wird die analytische Struktur des Quarkpropagators in der komplexen Ebene numerisch untersucht und die Möglichkeit getestet die Anwendbarkeit auf den Sektor der schwer-leicht Quark Systeme im skalaren und pseudoskalaren Kanal, wie dem D Meson, durch Variation des Impulsteilungsparameters zu erweitern. Die Lösungen der Dyson-Schwinger-Gleichung in der Wigner-Weyl-Phase der chiralen Symmetrie bei nichtverschwindenden Stromquarkmassen wird benutzt um den Fall einer expliziten Symmetriebrechung ohne spontane Symmetriebrechung zu untersuchen. Abstract The interplay of hadron properties and their modification in an ambient nuclear medium on the one hand and spontaneous chiral symmetry breaking and its restoration on the other hand is investigated. QCD sum rules for D and B mesons embedded in cold nuclear matter are evaluated. We quantify the mass splitting of D D¯ and B B¯ mesons as a function of the nuclear matter density and investigate the − − ∗ impact of various condensates in linear density approximation. The analysis also includes Ds and D0 mesons. QCD sum rules for chiral partners in the open-charm meson sector are presented at nonzero baryon net density or temperature. We focus on the differences between pseudo-scalar and scalar as well as vector and axial-vector D mesons and derive the corresponding Weinberg type sum rules. Based on QCD sum rules we explore the consequences of a scenario for the ρ meson, where the chiral symmetry breaking condensates are set to zero whereas the chirally symmetric condensates remain at their vacuum values. The complementarity of mass shift and broadening is discussed. An alternative approach which utilizes coupled Dyson-Schwinger and Bethe-Salpter equations for quark-antiquark bound states is investigated. For this purpose we analyze the analytic structure of the quark propagators in the complex plane numerically and test the possibility to widen the applicability of the method to the sector of heavy-light mesons in the scalar and pseudo-scalar channels, such as the D mesons, by varying the momentum partitioning parameter. The solutions of the Dyson-Schwinger equation in the Wigner-Weyl phase of chiral symmetry at nonzero bare quark masses are used to investigate a scenario with explicit but without dynamical chiral symmetry breaking. iv Contents Kurzdarstellung iii Abstract iii 1 Introduction and motivation 1 1.1 Dynamical chiral symmetry breaking . .3 1.2 Hadrons in the medium . .5 1.3 QCD sum rules . .7 1.4 Bethe-Salpeter–Dyson-Schwinger approach . 10 1.5 Experimental and theoretical status and perspectives . 12 1.6 Structure of the thesis . 14 2 Introduction to QCD sum rules 17 2.1 Subtracted dispersion relations . 21 2.2 Operator product expansion . 23 2.3 Sum rules . 28 3 QCD sum rules for heavy-light mesons 31 3.1 Operator product expansion . 31 3.2 Parameterizing the spectral function . 32 3.3 Evaluation for D mesons . 35 3.3.1 The pseudo-scalar case . 35 3.3.2 The scalar case . 41 3.4 Evaluation for B mesons . 44 3.5 Interim summary . 45 4 Chiral partner QCD sum rules 47 4.1 Differences of current-current correlators and their OPE . 48 4.2 Chiral partners of open charm mesons . 53 4.2.1 The case of P–S . 53 4.2.2 The case of V–A . 55 4.2.3 Heavy-quark symmetry . 56 4.2.4 Numerical examples . 56 v 4.3 Interim summary . 58 5 VOC scenario for the ρ meson 59 5.1 Chiral transformations and QCD condensates . 61 5.2 VOC scenario for the ρ meson . 64 5.3 Mass shift vs. broadening . 67 5.4 Notes on ! and axial-vector mesons . 77 5.5 Interim summary . 78 6 Introduction to Dyson-Schwinger and Bethe-Salpeter equations 81 6.1 Dyson-Schwinger equations . 81 6.1.1 Derivation . 81 6.1.2 Renormalization . 84 6.1.3 Rainbow approximation . 84 6.2 Bethe-Salpeter equations . 85 6.2.1 Derivation . 86 6.2.2 Ladder approximation . 88 6.2.3 Solving the Bethe-Salpeter equation in ladder approximation . 90 7 Dyson-Schwinger and Bethe-Salpeter approach 93 7.1 DSE for the quark propagator . 93 7.1.1 Analytical angle integration of the infrared part of the potential . 96 7.1.2 Solution along the real axis . 98 7.1.3 Solution in the complex plane . 99 7.2 Bethe-Salpeter equation for mesons . 108 7.3 Wigner-Weyl solution . 115 8 Summary and outlook 121 Appendices 127 A Chiral symmetry, currents and order parameters 127 A.1 The formalism for classical fields . 128 A.2 The formalism for quantized fields . 142 A.3 Brief survey on Quantum Chromodynamics . 150 B Correlation functions 155 B.1 Källén-Lehmann representation and analytic properties . 156 B.2 Symmetry constraints . 159 B.3 Decomposition . 163 B.4 Non-anomalous Ward identities . 165 vi B.5 Subtracted dispersion relations . 170 B.5.1 Vacuum dispersion relations . 171 B.5.2 In-medium dispersion relations . 173 C In-medium OPE for heavy-light mesons 177 C.1 Fock-Schwinger gauge and background field method . 177 C.2 Borel transformed sum rules . 183 C.3 Projection of color, Dirac and Lorentz indices . 188 C.4 OPE for the D meson . 191 C.5 Absorption of divergences . 196 C.6 Recurrence relations . 199 C.7 Sum rule analysis for heavy-light mesons . 205 C.8 Condensates near the deconfinement transition . 207 D Chiral partner QCD sum rules addendum 211 D.1 Dirac projection . 211 D.2 The perturbative quark propagator . 212 D.3 Cancellation of IR divergences . 213 Acronyms 217 List of Figures 219 List of Tables 223 Bibliography 225 vii 1 Introduction and motivation Within the present standard model of particle physics the building blocks of matter are quarks and leptons. These fermions interact via gravity, as well as weak, electromagnetic and strong interaction - the four known forces in nature. Apart from gravity, these interactions may be described by virtue of gauge field theories, where the interaction is transferred by gauge bosons. One such theory is quantum chromodynamics (QCD) which is believed to describe the strong interaction. The gauge bosons of QCD are called gluons. At the time when this thesis has been written, the last missing piece of the standard model is the Higgs boson, which is predicted in order to explain the masses of the gauge bosons carrying the weak interaction force. Moreover, the interaction of the Higgs boson with the fermion fields within the standard model also gives rise to the bare masses of the quarks and some of the leptons. Recently, first results of the Higgs boson search from the ATLAS and ALICE experiments at the Large Hadron Collider at CERN have been presented [ATL, ALI], and physicists around the world are excitingly awaiting the final announcements. However, the main contribution to the mass of matter that surrounds us has a different origin. The mass of “usual matter” resides in nuclei composed of nucleons, i.e. protons and neutrons. Nucleons belong to the hadrons and the fundamental degrees of freedom within an environment below the deconfinement transition of QCD (to be explained later on) are hadrons. Hadrons are strongly interacting, subatomic states of quarks and gluons, which interact via and form bound states by virtue of the strong interaction. They may be cataloged by their spin. Bosonic hadrons are called mesons and consist of an even number of valence quarks. Fermionic hadrons are called baryons and consist of an odd number of valence quarks, see Fig. 1.0.1. Valence quarks are the quarks which determine the quantum numbers of the hadron. The bare masses of quarks are well known from several experiments and gluons are supposed to be massless due to the exact local SUc(3) gauge symmetry. But against naive intuition, summing up all constituent masses of a hadron, one can not explain its mass, which would be expected to be smaller than the sum of the constituent masses due to the binding energy. In contrast, the hadron masses are much larger than the sum of their constituents masses. Let us consider, for example, the nucleon. In the naive quark model it consists of three light quarks with the sum of their bare current quark masses as obtained from several high energy experiments being less than 20 MeV, while the nucleon mass is about 938 MeV[Nak10].
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