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| A Bayesian Analysis of QCD Sum Rules |
PREFACE :
The author develops a novel analysis method for QCD sum rules (QCDSR) by applying the maximum entropy method (MEM) to arrive at an analysis with less artificial assumptions than previously held. This is a first-time accomplishment in the field. In this thesis, a reformed MEM for QCDSR is formalized and is applied to the sum rules of several channels: the light-quark meson in the vector channel, the light-quark baryon channel with spin and isospin 1/2, and several quarkonium channels at both zero and finite temperatures. This novel technique of combining QCDSR with MEM is applied to the study of quarkonium in hot matter, which is an important probe of the quark-gluon plasma currently being created in heavy-ion collision experiments at RHIC and LHC.
Table of Contents
Part I Introduction and Review
Chapter 1 Introduction
1.1 Describing Hadrons from QCD
1.2 QCD Sum Rules and Its Ambiguities
1.3 Hadrons in a Hot and/or Dense Environment
1.4 Motivation and Purpose of this Thesis
1.5 Outline of the Thesis
Chapter 2 Basic Properties of QCD
2.1 The QCD Lagrangian
2.2 Asymptotic Freedom
2.3 Symmetries of QCD 2.3.1 Gauge Symmetry
o 2.3.2 Chiral Symmetry
o 2.3.3 Dilatational Symmetry
o 2.3.4 Center Symmetry
2.4 Phases of QCD
Chapter 3 Basics of QCD Sum Rules
3.1 Introduction
o 3.1.1 The Theoretical Side
o 3.1.2 The Phenomenological Side
o 3.1.3 Practical Versions of the Sum Rules
3.2 More on the Operator Product Expansion
o 3.2.1 Theoretical Foundations
o 3.2.2 Calculation of Wilson Coefficient
3.3 More on the QCD Vacuum
o 3.3.1 The Quark Condensate
o 3.3.2 The Gluon Condensate
o 3.3.3 The Mixed Condensate
o 3.3.4 Higher Order Condensates
3.4 Parity Projection for Baryonic Sum Rules
o 3.4.1 The Problem of Parity Projection in Baryonic Sum Rules
o 3.4.2 Use of the "Old Fashioned" Correlator
o 3.4.3 Construction of the Sum Rules
o 3.4.4 General Analysis of the Sum Rules for Three-Quark Baryons
Chapter 4 The Maximum Entropy Method
4.1 Basic Concepts
o 4.1.1 The Likelihood Function and the Prior Probability
o 4.1.2 The Numerical Analysis
o 4.1.3 Error Estimation
4.2 Sample MEM Analysis of a Toy Model
o 4.2.1 Construction of the Sum Rules
o 4.2.2 MEM Analysis of the Borel Sum Rules
o 4.2.3 MEM Analysis of the Gaussian Sum Rules
o 4.2.4 Summary of Toy Model Analysis
Part II Applications
Chapter 5 MEM Analysis of the . Meson Sum Rule
5.1 Introduction
5.2 Analysis Using Mock Data
o 5.2.1 Generating Mock Data and the Corresponding Errors
o 5.2.2 Choice of an Appropriate Default Model
o 5.2.3 Investigation of the Stability of the Obtained Spectral Function
o 5.2.4 Estimation of the Precision of the Final Results
o 5.2.5 Why it is Difficul to Accurately Determine the Width of the . Meson
5.3 Analysis Using the OPE Results 5.3.1 The . Meson Sum Rule
o 5.3.2 Results of the MEM Analysis
5.4 Summary and Conclusion
Chapter 6 MEM Analysis of the Nucleon Sum Rule
6.1 Introduction
6.2 QCD Sum Rules for the Nucleon
o 6.2.1 Borel Sum Rule
o 6.2.2 Gaussian Sum Rule
6.3 Analysis Using the Borel Sum Rule
o 6.3.1 Analysis Using Mock Data
o 6.3.2 Analysis Using OPE Data
6.4 Analysis Using the Gaussian Sum Rule
o 6.4.1 Analysis Using Mock Data
o 6.4.2 Analysis Using OPE Data
o 6.4.3 Investigation of the � Dependence
6.5 Summary and Conclusion
Chapter 7 Quarkonium Spectra at Finite Temperature from QCD Sum Rules and MEM
7.1 Introduction
7.2 Formalism
o 7.2.1 Formulation of the Sum Rule
o 7.2.2 The Temperature Dependence of the Condensates
7.3 Results of the MEM Analysis for Charmonium 7.3.1 Mock Data Analysis
o 7.3.2 OPE Analysis at T= 0
o 7.3.3 OPE Analysis at T = 0
o 7.3.4 Summary for Charmonium
7.4 Results of the MEM Analysis for Bottomonium
o 7.4.1 Mock Data Analysis
o 7.4.2 OPE Analysis at T= 0
o 7.4.3 OPE Analysis at T = 0
o 7.4.4 Summary for Bottomonium
Part III Concluding Remarks
Chapter 8 Summary, Conclusion and Outlook
8.1 Summary and Conclusion
8.2 Outlook
Appendix A The Dispersion Relation
Appendix B The Fock-Schwinger Gauge
Appendix C The Quark Propagator
Appendix D Non-Perturbative Coupling of Quarks and Gluons
Appendix E Gamma Matrix Algebra
Appendix F The Fourier Transformation
Appendix G Derivation of the Shannon-Jaynes Entropy
Appendix H Uniqueness of the Maximum of P[.|GH]
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