Numerical Techniques for High-Precision Multi-Loop Calculations in Quantum Electrodynamics and Scalar Field Theories

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Multi-loop calculations in quantum field theory, especially quantum electrodynamics, have become more and more important along the years because many high-precision experiments and determinations of fundamental constants rely on predictions which can be extracted from these powerful theoretical tools. Renormalized perturbative field theory forms the basis for the most accurately understood systems and physical properties in nature. In this proposal, we focus on general numerical techniques for the evaluation of multi-loop diagrams, with numerous potential applications, and on specific planned calculations in various cases of particular interest. These are related to rather urgent aspects of higher-order binding corrections to the Lamb shift and to the bound-electron g-factor, and on the perturbative renormalization group treatment of critical phenomena like the superfluidity of helium, where a recent experiment has produced a value which is more accurate than the best available theory.