Exposing the threshold structure of loop integrals

Abstract

The understanding of the physical laws determining the infrared behaviour 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 non-simplicial 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. Contrary to the TOPT and LTD representations, 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 classify infrared-finite scattering theories, numerically evaluate loop integrals and to simplify threshold regularisation procedures.

7 pages

References in corpus (12)

• Path integral approach to eikonal and next-to-eikonal exponentiation
• On gravitational description of Wilson lines
• Geometric approach to asymptotic expansion of Feynman integrals
• From loops to trees by-passing Feynman's theorem
• Webs in multiparton scattering using the replica trick
• Four-gluon scattering at three loops, infrared structure and Regge limit
• A Tree-Loop Duality Relation at Two Loops and Beyond
• Numerical NLO QCD calculations
• Infrared quantum information
• Ultraviolet divergences and factorization for coordinate-space amplitudes
• On the non-cancellation of infrared singularities in collisions of massive quarks
• Flow-oriented perturbation theory

Cited by in corpus (6)

• Light-by-Light Scattering at Next-to-Leading Order in QCD and QED
• Local infrared safety in time-ordered perturbation theory
• Locally finite two-loop amplitudes for electroweak production through gluon fusion
• Tropical sampling from Feynman measures
• NNLO QCD corrections to $\gamma \gamma \to Q\bar{Q}$ from Local Unitarity combined with Coulomb resummation and NLO EW effects
• Coordinate light-cone-ordered perturbation theory