Some Remarks on Non-Planar Diagrams

Two criteria for planarity of a Feynman diagram upon its propagators (momentum flows) are presented. Instructive Mathematica programs that solve the problem and examples are provided. A simple geometric argument is used to show that while one can planarize non-planar graphs by embedding them on higher-genus surfaces (in the example it is a torus), there is still a problem with defining appropriate dual variables since the corresponding faces of the graph are absorbed by torus generators.

Reductions and Contractions of 1-loop Tensor Feynman Integrals

We report on the progress in constructing contracted one-loop tensors. Analytic results for rank R=4 tensors, cross-checked numerically, are presented for the first time.