Exposing the threshold structure of loop integrals

The understanding of the physical laws determining the infrared behavior of amplitudes is a longstanding and topical problem. In this paper, we show that energy conservation alone implies strong constraints on the threshold singularity structure of Feynman diagrams. In particular, we show that it implies a representation of loop integrals in terms of Fourier transforms of nonsimplicial convex cones. We then engineer a triangulation that has a direct diagrammatic interpretation in terms of a straightforward edge-contraction operation. We use it to develop an algorithmic procedure that performs the Fourier integrations in closed form, yielding the novel cross-free family three-dimensional representation of loop integrals. Its singularity structure is entirely and elegantly expressed in terms of the graph-theoretic notions of connectedness and crossing. These results can be used to study the Kinoshita-Lee-Nauenberg cancellation mechanism, numerically evaluate loop integrals and to simplify threshold regularization procedures.

Automated summary

In this paper, the authors investigate the infrared behavior of amplitudes and show that energy conservation imposes strong constraints on the threshold singularity structure of Feynman diagrams. They develop an algorithmic procedure using Fourier transforms and graph-theoretic notions, which enables closed-form Fourier integrations and introduces a novel three-dimensional representation of loop integrals. These results have implications for studying cancellation mechanisms, evaluating loop integrals numerically, and simplifying threshold regularization procedures.