Loop-tree duality from vertices and edges

The causal representation of multi-loop scattering amplitudes, obtained from the application of the loop-tree duality formalism, comprehensively elucidates, at integrand level, the behaviour of only physical singularities. This representation is found to manifest compact expressions for multi-loop topologies that have the same number of vertices. Interestingly, integrands considered in former studies, with up-to six vertices and L internal lines, display the same structure of up-to four-loop ones. The former is an insight that there should be a correspondence between vertices and the collection of internal lines, edges, that characterise a multi-loop topology. By virtue of this relation, in this paper, we embrace an approach to properly classify multi-loop topologies according to vertices and edges. Differently from former studies, we consider the most general topologies, by connecting vertices and edges in all possible ways. Likewise, we provide a procedure to generate causal representation of multi-loop topologies by considering the structure of causal propagators. Explicit causal representations of loop topologies with up-to nine vertices are provided.

Automated summary

The causal representation of multi-loop scattering amplitudes obtained from the loop-tree duality formalism provides a comprehensive understanding of the behavior of physical singularities at the integrand level. The study reveals compact expressions for multi-loop topologies with the same number of vertices and shows a correspondence between vertices and the collection of internal lines. By considering the structure of causal propagators, the paper presents explicit causal representations of loop topologies with up to nine vertices.