Loop-tree duality from vertices and edges

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 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.

Four-loop scattering amplitudes journey into the forest

“…A causal representation of the multiloop N 3 MLT was analytically reconstructed in Ref. [27,28,36] by matching all the combinations of four thresholds that are causally compatible to each other. There are thirteen causal denominators which represent potential singular configurations and they are constructed from sums of on-shell energies exclusively, i.e.…”

A Stroll through the Loop-Tree Duality

“…In particular, it turned out to be an excellent tool to unveil the structure of causal singularities of multi-loop multi-leg Feynman integrals and scattering amplitudes. There have been very recent findings regarding general all-loop formulae to describe any N k MLT amplitude, using clever algebraic relations among them [100]. This computational technology has been implemented in the package LOTTY [101], which allows automatic obtaining of the causal representation of multi-loop Feynman integrals and scattering amplitudes.…”

“…Describing diagrams in terms of vertices (i.e., interaction vertices) and edges (i.e., sets of propagators connecting two fixed vertices) was also proposed in Ref. [100]. This description leads to the vertex matrix, a geometrical object that allows one to completely characterize the causal structure of a given diagram.…”

“…In fact, an understanding of the structure of the latter started to appear in the literature by means of geometric properties [102] and is yet under consideration. On top of the former approach, it was recently conjectured a breathtaking formulation of the all-loop causal representation of multi-loop scalar integrands, obtained from the features that characterized a topology, vertices and edges [100]. This formulation provides, regardless of the loop order, the most symmetric causal representation of Feynman integrands, in a complementary and independent way to the nested residue calculation used by the LTD formalism.…”

“…This was applied to remove unphysical threshold singularities and obtain causal integrand-level representations of several topological families of multi-loop multi-leg Feynman integrals [92][93][94][95][96][97][98][99]. Moreover, all-order causal formulae were obtained using novel algebraic relations [100], leading to an automatized framework to implement these calculations [101]. A purely geometrical interpretation [102] was also developed, showing a complementary approach to tackle the causal structure of multi-loop multi-leg scattering amplitudes and Feynman integrals.…”