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In the past years, we have been developing a novel technique, calledFour-Dimensional Unsubtraction (FDU) which aims to obtain purelyfour-dimensional representations of the matrix elements contributing tophysical observables. In this talk, we describe the application of theloop-tree duality (LTD) theorem to represent loop amplitudes in terms oftree-level like objects, focusing on the origin of possible singularities ofscattering amplitudes. In particular, we analyze the regions responsible ofinfrared and threshold singularities. With this information, we aim to extendthe FDU formalism to NNLO and beyond.

Analysis of singularities and the four-dimensional representation of physical observables within the LTD formalism