An analyst’s incursion into quantum field theory

In quantum field theory, scattering amplitudes are described by sums over terms described by Feynman diagrams.  In many cases, these are meant to describe singular divergent integrals, so it is not clear how they should be interpreted and whether different interpretations are equivalent.  In the Euclidean context, we turn this into a clean problem of real analysis and show how algebraic techniques allow to tackle it.