Computational and mathematical aspects of Feynman integrals

Abstract

This thesis covers a number of different research projects which are all connected to the central topic of computing Feynman integrals efficiently through analytic methods. Improvements in our ability to evaluate Feynman integrals allow us to increase the order in perturbation theory at which we are able to produce theoretical predictions for various processes in the Standard Model, which can be tested at the Large Hadron Collider. In the first part of this thesis, we cover novel research on the analytic computation of elliptic Feynman integrals. We will show how certain elliptic Feynman integrals can be written as one-fold integrals over polylogarithmic Feynman integrals, which can be solved from systems of differential equations in a canonical dlog-form, or by using the method of direct integration. Thereafter, we discuss a method for computing Feynman integrals from their differential equations in terms of one-dimensional series expansions along contours in phase-space. By connecting series expansions along multiple line segments, the method allows us to obtain high precision numerical results for various Feynman integrals at arbitrary points in phase-space. We will also present a novel Mathematica package called DiffExp, which provides a general implementation of these series expansion methods. As an illustrative example, we apply the package to obtain high precision results for the unequal-mass banana graph family in the physical region. Next, we present the computation of the complete set of non-planar master integrals relevant for Higgs plus jet production at next-to-leading order with full heavy quark mass dependence. The non-planar integrals fit into two integral families. We provide a choice of basis that puts many of the sectors in a canonical dlog-form, and we show that high precision numerical results can be obtained for all integrals using series expansion methods. Lastly, we discuss work on the diagrammatic coaction of the equal-mass elliptic sunrise family.