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 (31)

• 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
• Celestial Amplitudes from UV to IR
• 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
• Landau Discriminants
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Tropical Monte Carlo quadrature for Feynman integrals
• Infrared Divergences and the Eikonal Exponentiation
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• Steinmann Relations and the Wavefunction of the Universe
• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
• Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order
• What is the $i\epsilon$ for the S-matrix?
• Non-abelian infrared divergences on the celestial sphere
• Geometrical approach to causality in multi-loop amplitudes
• Ultraviolet divergences and factorization for coordinate-space amplitudes
• Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation
• Loop-tree duality from vertices and edges
• Numerical integration of loop integrals through local cancellation of threshold singularities
• Universal opening of four-loop scattering amplitudes to trees
• On the non-cancellation of infrared singularities in collisions of massive quarks
• Cutting rules on a cylinder: a bottom-up approach to quantum kinetic theory for the early universe
• Graph complexes and Feynman rules
• Flow-oriented perturbation theory
• Algebraic Interplay between Renormalization and Monodromy

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• 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