Feynman Integral Calculus and Hypergeometric Functions: New Results, Interconnections and Computer Algebra

Abstract

Feynman integrals play a fundamental role in perturbative calculations within the realm of quantum field theory. In this thesis, we study several important mathematical properties of multi-loop multi-leg Feynman Integrals in dimensional regularization and their realization in terms of well-known functions of mathematical physics, especially hypergeometric functions, including in their Laurent expansion in the dimensional parameter epsilon. We study the GKZ systems using the Gr\"obner deformation technique, which offers solutions for these integrals expressed in terms of hypergeometric functions with multiple variables. We implement this algorithm as a \textsc{Mathematica} package, which has become part of the publication \texttt{FeynGKZ.wl}. It is important to note that multivariate hypergeometric functions exhibit their validity within specific domains of convergence, which can be obtained by Horn's theorem. To extend our analysis beyond these defined domains and determine numerical values, the pursuit of analytic continuations becomes imperative. We explore the method of Olsson for this purpose, developing it into the \textsc{Mathematica} package\texttt{ Olsson.wl}, thereby streamlining the process of calculating these analytic continuations. In addition to this, we introduce the \texttt{ROC2.wl} package, which serves as a realization of Horn's theorem, capable of finding the domains of convergence for bi-variate hypergeometric functions. Our exploration further extends to the discovery of analytic continuations for Appell $F_2$ and Horn $H_5$ functions through the same method. We optimize the analytic continuations of Appell $F_2$ to facilitate numerical evaluation for generic values of parameters and for real values of its arguments except for its singular curves, giving rise to the foundational package, \texttt{AppellF2.wl}. Lastly, we present an algorithm designed to determine the series expansion of hypergeometric functions based on their series representation. This algorithm is implemented in the \textsc{Mathematica} package \texttt{MultiHypExp.wl}, offering a tool to expand certain hypergeometric functions in series around integer-valued parameters in terms of multiple polylogarithms. Our works demonstrate the importance of the impact of theory and computer algebra on modern Feynman Integral calculus and provide general-purpose open-source packages which are bound to be of great utility to the community.