Generalised Unitarity, Integrand Decomposition, and Hidden properties of QCD Scattering Amplitudes in Dimensional Regularisation

In this thesis, we present new developments for the analytic calculation of tree- and multi-loop level amplitudes. Similarly, we study and extend their analytic properties. We propose a Four-dimensional formulation (FDF) equivalent to the four-dimensional helicity scheme (FDH). In our formulation, particles propagating inside the loop are represented by four dimensional massive internal states regulating the divergences. We provide explicit four-dimensional representations of the polarisation and helicity states of the particles propagating in the loop. Within FDF, we use integrand reduction and four dimensional unitarity to perform analytic computations of one-loop scattering amplitudes. The calculation of tree level scattering amplitude, in this framework, allows for a simultaneous computation of cut-constructible and rational parts of one-loop scattering amplitudes. We present a set of non-trivial examples, showing that FDF scheme is suitable for computing important $2\to2,3,4$ partonic amplitudes at one-loop level. We start by considering two gluons production by quark anti-quark annihilation. Then, the (up to four) gluon production, $gg\to ng$ with $n=2,3,4$. And finally, the Higgs and (up to three) gluons production via gluon fusion, $gg\to ng\,H$ with $n=1,2,3$, in the heavy top mass limit. We also investigate, by following a diagrammatic approach, the role of colour-kinematics (C/K) duality of off-shell diagrams in gauge theories coupled to matter. We study the behaviour of C/K-duality for theories in four- and in $d$-dimensions. The latter follows the prescriptions given by FDF. We show that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a C/K-violating term due to the contributions of sub-graphs only. We discuss the role of the off-shell decomposition in the direct construction of higher-multiplicity numerators satisfying C/K-duality. We present the QCD process $gg\to q\bar{q}g$. An analogous study, within FDF, is carried out for $d$-dimensionally regulated amplitudes. The computation of dual numerators generates, as byproduct, relations between tree-level amplitudes with different orderings. These relations turn to be the Bern-Carrasco-Johansson (BCJ) identities for four- and $d$-dimensionally regulated amplitudes. We combine BCJ identities and integrand reduction methods to establish relations between one-loop integral coefficients for dimensionally regulated QCD amplitudes. We also elaborate on the radiative behaviour of tree-level scattering amplitudes in the soft regime. We show that the subleading soft term in single-gluon emission of quark-gluon amplitudes in QCD is controlled by differential operators, whose universal form can be derived from both Britto-Cachazo-Feng-Witten recursive relations and gauge invariance, as it was shown to hold for graviton and gluon scattering. In the last part of the thesis, we describe the main features of the multi-loop calculations. We briefly describe the adaptive integrand decomposition (AID), a variant of the standard integrand reduction algorithm. AID exploits the decomposition of the space-time dimension in parallel and orthogonal subspaces. We focus, in particular, on the calculation of $2\to2,3$ partonic amplitudes at two loop-leve