Scattering Amplitudes and Logarithmic Differential Forms

Abstract

This thesis is about the analytical computation of Feynman integrals and scattering amplitudes in quantum field theory. The topics of this thesis can be grouped into three categories: development of algorithms, five-particle scattering, and infrared divergences. The two algorithms we implemented automate key steps of the computation of Feynman integrals and scattering amplitudes, which previously required a large amount of manual and heuristic labor. With the first algorithm we classify Feynman integrals with particular analytic properties, namely those whose integrands can be expressed in terms of so-called dlog forms. Feynman integrals of this special type are particularly easy to compute using differential equations. This algorithm is of central importance for all applications in this thesis. With the second algorithm we address a frequent obstruction in analytic computa tions which is the emergence of square roots in otherwise rational expressions. The algorithm searches for reparametrizations of these expressions such that all square roots cancel out and hence the computation simplifies significantly. With the help of the first algorithm we analytically computed Feynman inte grals with up to four loops and up to five particles. We used these integrals to compute, for the first time, full five-particle scattering amplitudes at two loop order in N=4 super Yang-Mills theory, N=8 supergravity, and quantum chromodynamics (QCD). These results are important to investigate the math ematical structures of the different quantum field theories which are especially rich for the supersymmetric theories under consideration. The QCD result can be seen as the starting point for the computation of further scattering amplitudes that are highly relevant for phenomenology such as three-jet production at next-to-next-to-leading order in perturbation theory. Finally, we studied infrared divergences in the context of a four-loop form factor computation, where we computed a particularly important part of the light-like cusp anomalous dimension. An essential part of this calculation was the systematic analysis of the infrared divergences for the Feynman integrals involved.