Scattering Amplitudes in Gauge Theories

This thesis is focused on the development of new mathematical methods for computing multi-loop scattering amplitudes in gauge theories. In this work we combine, for the first time, the unitarity-based construction for integrands, and the recently introduced integrand-reduction through multivariate polynomial division. After discussing the generic features of this novel reduction algorithm, we will apply it to the one- and two-loop five-point amplitudes in ${\cal N}=4$ sYM. The integrands of the multiple-cuts are generated from products of tree-level amplitudes within the super-amplitudes formalism. The corresponding expressions will be used for the analytic reconstruction of the polynomial residues. Their parametric form is known a priori, as derived by means of successive polynomial divisions using the Gr\"obner basis associated to the on-shell denominators. The integrand reduction method will be exploited to investigate the color-kinematic duality for multi-loop ${\cal N}=4$ sYM scattering amplitudes. Our analysis yields a suggestive, systematic way to generate graphs which automatically satisfy the color-kinematic dualities. Finally, we will extract the leading ultra-violet divergences of five-point one- and two-loop amplitudes in ${\cal N}=4$ sYM, which represent a paradigmatic example for studying the UV behavior of supersymmetric amplitudes.