Mathematical aspects of Feynman integrals [Elektronische Ressource] / Christian Bogner

Mathematical Aspects of Feynman IntegralsDissertationzur Erlangung des Grades“Doktor der Naturwissenschaften”am Fachbereich Physik¨der Johannes Gutenberg-Universitatin MainzChristian Bognergeboren in MainzMainz, August 200923“What can be said at all can be said clearly; and whereof one cannot speakthereof one must be silent.”(Ludwig Wittgenstein)ContentsChapter 1. Introduction 71.1. General Introduction 71.2. Outline 8Chapter 2. Preliminaries on Feynman Integrals 112.1. Theories, Graphs and Rules 112.2. Dimensional Regularization 202.3. Feynman Parameters 242.4. From Tensors to Scalars 28Chapter 3. Integration-by-Parts for Feynman Integrals 313.1. The IBP-Approach 323.2. Translation to Feynman Parameters 41Chapter 4. Feynman Graph Polynomials 494.1. Graphs, Feynman Graphs and Associated Matrices 504.2. Symanzik Polynomials and Theorems of the Matrix-Tree-Type 644.3. The Multivariate Tutte Polynomial 73Chapter 5. Sector Decomposition and Resolution of Singularities 795.1. Sector Decomposition 805.2. Hironaka’s Polyhedra Game 965.3. Resolution of Singularities 107Chapter 6. Periods 1176.1. Periods, Nested Sums and Iterated Integrals 1186.2. Periods and Feynman Integrals 1306.3. Outlook on Periods and Feynman Integrals 135Chapter 7.