Abstract
The computation of multi-loop multi-leg scattering amplitudes plays a key role to improve the precision of theoretical predictions for particle physics at high-energy colliders. In this work, we focus on the mathematical properties of the novel integrand-level representation of Feynman integrals, which is based on the Loop-Tree Duality (LTD). We explore the behaviour of the multi-loop iterated residues and explicitly show, by developing a general compact and elegant proof, that contributions associated to displaced poles are cancelled out. The remaining residues, called nested residues as originally introduced in ref. [1], encode the relevant physical information and are naturally mapped onto physical configurations associated to nondisjoint on-shell states. By going further on the mathematical structure of the nested residues, we prove that unphysical singularities vanish, and show how the final expressions can be written by using only causal denominators. In this way, we provide a mathematical proof for the all-loop formulae presented in ref. [2].
Highlights
Involves dealing with multi-loop multi-leg Feynman integrals
We focus on the mathematical properties of the novel integrand-level representation of Feynman integrals, which is based on the Loop-Tree Duality (LTD)
On top of the studies of decomposition at integrand level, different representations of Feynman integrals [39, 40] promoted the use of algebraic geometry [41,42,43,44,45,46] and intersection theory [47,48,49,50] to perform a reduction of a multi-loop amplitude to master integrals
Summary
We establish the mathematical basis of the iterated residue approach for the multi-loop dual representation in the context of the Loop-Tree Duality formalism. To begin the discussion, we will consider variables, functions and their pole structure, trying to identify the properties of the multi-variable residue. For this reason, we introduce the physical concepts from the mathematical formalism, in order to appreciate the generality of the presentation. The primitive variables are extended to the complex plane successively, but not simultaneously, which implies that when computing the residue for xi, we promote xi ∈ C but we keep xj ∈ R for any i = j;. It is mandatory to say that the residue is well defined because the arguments of the function contain only one complex variable, xi In this way, the iterated residue algorithm can be understood as an iterated application of the functor, Res ◦ i : CRL → CRL−1 ,. We will explicitly show that the integral in eq (2.4) can be computed by just looking at the residues of specific poles
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