Geometrical approach to causality in multi-loop amplitudes

TL;DR

This paper introduces a geometric graph theory-based method to analyze the causal structure of multi-loop scattering amplitudes in quantum field theory, improving understanding and computation of Feynman integrals.

Contribution

It presents a novel geometric approach that uses the vertex matrix to determine causal structures, aligning with Cutkosky rules and applicable across topologies.

Findings

Causal structure is fully determined by the vertex matrix.

Causal representations are obtained by summing over causal entangled thresholds.

Diagrams with the same vertices and edges have similar causal structures.

Abstract

An impressive effort is being placed in order to develop new strategies that allow an efficient computation of multi-loop multi-leg Feynman integrals and scattering amplitudes, with a particular emphasis on removing spurious singularities and numerical instabilities. In this article, we describe an innovative geometric approach based on graph theory to unveil the causal structure of any multi-loop multi-leg amplitude in Quantum Field Theory. Our purely geometric construction reproduces faithfully the manifestly causal integrand-level behaviour of the Loop-Tree Duality representation. We found that the causal structure is fully determined by the vertex matrix, through a suitable definition of connected partitions of the underlying diagrams. Causal representations for a given topological family are obtained by summing over subsets of all the possible causal entangled thresholds that originate connected and oriented partitions of the underlying topology. These results are compatible with Cutkosky rules. Moreover, we found that diagrams with the same number of vertices and multi-edges exhibit similar causal structures, regardless of the number of loops.