  We propose a novel representation of differential scattering cross-sections
that locally realises the direct cancellation of infrared singularities
exhibited by its so-called real-emission and virtual degrees of freedom. We
take advantage of the Loop-Tree Duality representation of each individual
forward-scattering diagram and we prove that the ensuing expression is locally
free of infrared divergences, applies at any perturbative order and for any
process without initial-state collinear singularities. Divergences for loop
momenta with large magnitudes are regulated using local ultraviolet
counterterms that reproduce the usual Lagrangian renormalisation procedure of
quantum field theories. Our representation is especially suited for a numerical
implementation and we demonstrate its practical potential by computing fully
numerically and without any IR counterterm the next-to-leading order accurate
differential cross-section for the process $e^+ e^- \rightarrow d \bar{d}$. We
also show first results beyond next-to-leading order by computing interference
terms part of the N4LO-accurate inclusive cross-section of a $1\rightarrow 2+X$
scalar scattering process.
