Mathematical properties of nested residues and their application to multi-loop scattering amplitudes

TL;DR

This paper investigates the mathematical structure of nested residues in multi-loop Feynman integrals using Loop-Tree Duality, proving cancellation of unphysical singularities and simplifying the expressions to causal denominators, advancing the theoretical understanding of scattering amplitudes.

Contribution

It provides a rigorous mathematical proof that unphysical singularities cancel in nested residues, supporting the all-loop formulae in multi-loop scattering amplitude calculations.

Findings

Unphysical singularities vanish in nested residues.

Residues encode physical configurations related to on-shell states.

Final expressions can be written using only causal denominators.

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 formal proof for the first time, that contributions associated to displaced poles are cancelled out. The remaining residues, called nested residues as originally introduced in Ref. \cite{Verdugo:2020kzh}, 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. \cite{Aguilera-Verdugo:2020kzc}.