Scattering Amplitudes and Logarithmic Differential Forms
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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.