Authors:	Klausen, René Pascal
Advisor:	Bogner, Christian
Title:	Hypergeometric feynman integrals
Online publication date:	5-Jan-2023
Year of first publication:	2023
Language:	english
Abstract:	In this thesis we will study Feynman integrals from the perspective of A-hypergeometric functions, a generalization of hypergeometric functions which goes back to Gelfand, Kapranov, Zelevinsky (GKZ) and their collaborators. This point of view was recently initiated by the works [74] and [150]. Inter alia, we want to provide here a concise summary of the mathematical foundations of A-hypergeometric theory in order to substantiate this viewpoint. This overview will concern aspects of polytopal geometry, multivariate discriminants as well as holonomic D-modules. As we will subsequently show, every scalar Feynman integral is an A-hypergeometric function. Furthermore, all coefficients of the Laurent expansion as appearing in dimensional and analytical regularization can be expressed by A-hypergeometric functions as well. By applying the results of GKZ we derive an explicit formula for series representations of Feynman integrals. Those series representations take the form of Horn hypergeometric functions and can be obtained for every regular triangulation of the Newton polytope Newt(U + F) of the sum of Symanzik polynomials. Those series can be of higher dimension, but converge fast for certain kinematical regions, which also allows an efficient numerical application. We will sketch an algorithmic approach which evaluates Feynman integrals numerically by means of these series representations. Further, we will examine possible issues which can arise in a practical usage of this approach and provide strategies to solve them. As an illustrative example we will present series representations for the fully massive sunset Feynman integral. Moreover, the A-hypergeometric theory enables us to give a mathematically rigorous description of the analytic structure of Feynman integrals (also known as Landau variety) by means of principal A-determinants and A-discriminants. This description of the singular locus will also comprise the various second-type singularities. Furthermore, we will find contributions to the singular locus occurring in higher loop diagrams, which seem to have been overlooked in previous approaches. By means of the Horn-Kapranov-parameterization we also provide a very efficient way to determine parameterizations of Landau varieties. We will illustrate those methods by determining the Landau variety of the dunce’s cap graph. We furthermore present a new approach to study the sheet structure of multivalued Feynman integrals by use of coamoebas.
DDC:	510 Mathematik 510 Mathematics 530 Physik 530 Physics
Institution:	Johannes Gutenberg-Universität Mainz
Department:	FB 08 Physik, Mathematik u. Informatik
Place:	Mainz
ROR:	https://ror.org/023b0x485
DOI:	http://doi.org/10.25358/openscience-8527
URN:	urn:nbn:de:hebis:77-openscience-7552b798-7cc3-48a6-b8bc-4d63c16ca5888
Version:	Original work
Publication type:	Dissertation
License:	CC BY-SA
Information on rights of use:	https://creativecommons.org/licenses/by-sa/4.0/
Extent:	viii, 206 Seiten ; Illustrationen, Diagramme
Appears in collections:	JGU-Publikationen