Loop-tree duality from vertices and edges

TL;DR

This paper develops a comprehensive classification and causal representation of multi-loop scattering amplitudes based on vertices and edges, extending previous work to more general topologies with up to nine vertices.

Contribution

It introduces a novel approach to classify multi-loop topologies using vertices and edges, and provides explicit causal representations for complex topologies.

Findings

Unified structure between vertices and edges in multi-loop topologies

Explicit causal representations for topologies with up to nine vertices

Generalized method for constructing causal representations of multi-loop diagrams

Abstract

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 \textit{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, \textit{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.