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.
Highlights
In light of providing an alternative approach to the numerical evaluation of multi-loop Feynman integrals, we elaborate on the loop-tree duality (LTD) formalism [47,48,49,50,51]
It has been observed that the application of the LTD formalism on multiloop Feynman integrands leads to a causal representation displaying only physical information [58, 59]
Inspired by the compact representation found in ref. [60], for the multi-loop Maximal (MLT), Next-to Maximal (NMLT) and Next-to-Next-to Maximal (N2MLT) loop topology configurations, it is desirable to understand the reason of the same structure of the causal representation, regardless of the loop order
Summary
We set the notation and recap the main features of the representation of the loop-tree duality (LTD) formalism by the application of nested residues. Let us remark that the dependence on the energy component of the loop momenta qi,0 is completely pulled out This is done, in order to apply the Cauchy residue theorem and integrate out one degree of freedom: the energy component of each loop momentum. Where the factor −2πı that comes from the Cauchy residue theorem is absorbed in the definition of the integration measure as shall be noted in the following. This residue corresponds to integrating out the energy components of the loop momenta. Is∈s where the iteration goes until the s-th set and corresponds to setting simultaneously L lines on shell The latter is equivalent to open the loop topology (or amplitude) into connected trees.
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