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Four-dimensional representation of scattering amplitudes and physical observables through the application of the Loop-Tree Duality theorem

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.