  In this thesis we propose a novel method to compute higher-order corrections
to physical cross sections, bypassing more traditional approaches. This
technique, the Four-Dimensional Unsubtraction (FDU), is based on the Loop-Tree
Duality (LTD) theorem, and aims at building pure four-dimensional
representations of scattering amplitudes at higher orders in perturbation
theory. This is done by locally renormalising the virtual contribution at high
energies, and then mapping the kinematics of the corresponding real
contribution so it matches the virtual one in the soft and collinear limits.
This makes the ultraviolet and infrared singularities vanish at the integrand
level when taking the sum of both contributions, meaning that the
four-dimensional limit can be taken before integrating, thus allowing for a
straightforward numerical implementation. We apply this method for the first
time to calculate the decay of a virtual photon to two massless quarks, and
then generalise the method to massive particles. We also show new advantages
provided by the LTD formalism. One of them is the possibility to write one- and
two-loop scattering amplitudes in a universal form at the integrand level,
regardless of the nature of the particle running inside the loop. The other is
to perform integrand-level asymptotic expansions in the internal mass - without
having to rely on expansion-by-region techniques - by taking advantage of the
Euclidean nature of the integration domain. Finally, we present a new algorithm
to locally renormalise two-loop amplitudes, where the renormalisation scheme
can be easily fixed through subleading contributions in the ultraviolet region.
