Geometrical approach to causality in multiloop amplitudes
Germán F. R. Sborlini
Abstract
An impressive effort is being pursued 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 behavior of the loop-tree duality representation. We find 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 find that diagrams with the same number of vertices and multi-edges exhibit similar causal structures, regardless of the number of loops.