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.
Highlights
Nowadays, one of the most successful descriptions of nature is based on quantum field theory (QFT)
We introduce the concept of maximal loop topology (MLT), which describes those diagrams or families of diagrams with the minimal number of sets for a given number of loops; i.e., n − L 1⁄4 1
We presented a geometrical study of the entangled causal structure of multi-loop multi-leg Feynman integrals and amplitudes
Summary
One of the most successful descriptions of nature is based on quantum field theory (QFT). Cutkosky rules [63,64] and Steinmann relations [65,66,67] establish a deep connection among geometrical properties of Feynman diagrams (cuts or partitions) and the structure of discontinuities of the underlying amplitude Inspired by these ideas, we investigated similar ideas with the purpose of reconstructing the whole amplitude at integrand level using a manifestly causal representation.
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