Some investigations into the methods and applications of modern Feynman Integral evaluation

Abstract

Feynman Integrals appear in perturbative quantum field theories and have been the focus of intensive study for over half a century. With the rise of the Standard Model of particle physics and its various extensions and the advent of high precision and high energy experiments, there has been a pressing need for the theory community to keep pace with experimentalists in their precision reach. Such precise theory predictions involve going to higher orders in the relevant coupling constant(s) in perturbation theory, where multi-loop and multi-leg Feynman Integrals appear.

In the first half of this thesis, we report our first investigation on certain quadratic and quartic integrals using one well-known method of obtaining series solutions to Feynman Integrals: the Method of Brackets. We then discuss some basic concepts from a systematic framework of analysing multi-variable hypergeometric functions, due to Gelfand, Kapranov and Zelevinsky (GKZ). These concepts include a geometrical approach based on computing triangulations of affine-space point configurations to obtain series solutions to what are known as GKZ hypergeometric functions. We discuss our implementation of this approach for obtaining series solutions to Feynman Integrals in the Mathematica code FeynGKZ and illustrate its use through a simple example: the one loop bubble Feynman Integral with two unequal masses.

The second half concerns three loop QCD corrections to the heavy-to-light form factors (HLFF). We compute full analytic results for the leading colour contributions, the complete light-quark contributions, and contributions from two heavy-quark loops. All relevant master integrals are calculated using the method of differential equations, and the results are expressed in terms of harmonic polylogarithms and generalised harmonic polylogarithms. We use the three loop results for the HLFF to consider the infra-red (IR) structure that emerges in their high-energy limit, and correspond it with the IR structures of the massless and massive form factors at three loops available in the literature.