Mathematical Aspects of Feynman Integrals

2 3 " What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent. " (Ludwig Wittgenstein) The idea that matter consists of smallest elements belongs to human culture since the ancient Greeks. The physics of the last few decades has given deep insight into the properties of these supposedly elementary particles. This illumination of the microcosmos is a story of success for two scientific concepts: the scattering experiment and quantum field theory. Scattering experiments, from Rutherford's scattering of α-particles off gold atoms in 1909 to the collision of highly accelerated protons at the Large Hadron Collider, expected to begin in 2009, have provided crucial information on the building blocks of matter. The success of a physical theory of elementary particles has to be judged by its capability to predict the values of quantities which can be measured in scattering experiments. The typical quantity to be measured in this kind of experiments is the so-called cross-section, from which in turn characteristic properties of the particles involved and their interaction can be deduced. The evolution of physical theories, probed on these experimental data, led to a model for the fundamental strong, electromagnetic and weak interactions of the known elementary particles. This so-called Standard Model is in good agreement with a wealth of observed phenomena. Nevertheless, the model is expected to be extended, depending on eagerly expected results from the Large Hadron Collider in the near future 1. The Standard Model is a highly successful quantum field theory whose application to the precise evaluation of an observable is usually far from trivial. In order to apply the model to the quantitative prediction of a cross-section one requires a perturbative formulation. The basic idea of perturbation theory is the assumption, that the interaction energies between the particles are relatively small compared to the energy of their free motion. The strength of an interaction is scaled by a so-called coupling parameter, which is assumed to be a relatively small quantity. An observable is then evaluated as a power series in the coupling parameter. This infinite series is truncated at a certain order and the precision of the result depends on the number of orders to be taken into account. The evaluation of the coefficients of this power series is in general highly elaborate. With increasing order of the perturbative expansion it becomes more and more …