A Subtraction Scheme for Feynman Integrals - Aaron Hillman (Princeton, Caltech)

Abstract: The infrared and ultraviolet divergence structure of Feynman integrals is a topic whose understanding is both of theoretical interest and practical necessity.  Recent work applying tropical geometry to the study of Feynman integrals has shed light on the infrared divergence structure of Feynman integrals and furnished new tools such as rapid numerical integration techniques for finite Feynman integrals. These crucially rely on the special properties of the geometries specific to Feynman integrals.  We extend this work to address the problem of extracting the full Laurent expansion in ε of UV divergent Feynman integrals in dimensional regularization, expressed only in terms of convergent integrals amenable to numerical integration. This is a problem with no canonical solution, which we solve by the introduction of u-variables, the analogue of cross-ratios (or dihedral coordinates) on the string world-sheet, for Feynman integrals. We comment on the prospects for an analogous solution in the case of IR divergent integrals.