Abstract
Elaborating on the novel formulation of the loop-tree duality, we introduce the Mathematica package Lotty that automates the latter at multi-loop level. By studying the features of Lotty and recalling former studies, we discuss that the representation of any multi-loop amplitude can be brought in a form, at integrand level, that only displays physical information, which we refer to as the causal representation of multi-loop Feynman integrands. In order to elucidate the role of Lotty in this automation, we recall results obtained for the calculation of the dual representation of integrands up-to four loops. Likewise, within Lotty framework, we provide support to the all-loop causal representation recently conjectured by the same author. The numerical stability of the integrands generated by Lotty is studied in two-loop planar and non-planar topologies, where a numerical integration is performed and compared with known results.
Highlights
Despite the plethora of diagrams that need to be considered when going to higher orders, new approaches, based on clever mathematical ideas, are currently being proposed to increase the efficiency and reduce the computing time in the several calculations [39,40,41,42,43,44,45,46]
In the spirit of harnessing from the knowledge of loop-tree duality (LTD), a novel formulation of the latter was recently provided in Ref. [83], where the approach to obtain a dual representation of multi-loop Feynman integrals, within the LTD framework, allowed for a complete automation and analysis regardless of the loop order, e.g. the works of Refs. [84,85,86,87] perform an analysis of up-to four loop topologies
In order to give a support to this conjecture, we implement in LOop-Tree dualiTY automation (Lotty) the close formula and give the procedure to generate the causal representation of any loop topology from its features
Summary
We set the notation for the multi-loop Feynman integrals and recap the main features of the novel formulation of the loop-tree duality (LTD) developed in Ref. [83]. Let us note that in the definition of Feynman propagators (4), we explicitly pulled out the dependence on the energy component of the loop momenta, qi,0. The latter is carried out in order to profit of the Cauchy residue theorem and integrate out one degree of freedom, which corresponds to the energy component of each loop momentum. I 1 ∈1 where the factor −2πı that comes from the Cauchy residue theorem is absorbed in the definition of the integration measure as shall be noted below As mentioned before, this residue corresponds to integrating out the energy components of the loop momenta and allows to introduce the nested residue as follows, A(DL)
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