Scattering equations formalism for QCD and supersymmetric Yang-Mills theories (2017)
Scattering amplitudes are one of the most important observables in perturbative quantum field theory, because they allow for the calculation of cross-sections, which are central to collision experiments. In this thesis, we perform a thorough review of some of the modern method for the calculation of tree-level amplitudes, the leading order contributions to the perturbative expansion of scattering amplitudes, focusing on the gauge theories that make up the standard model of particle physics. We will study methods for their calculation that overcome the issues and inefficiencies of Feynman diagrams, focusing on the Cachazo-He-Yuan (CHY) formalism, which provides closed formulas for tree amplitudes in arbitrary dimension as integrals over n-punctured Riemann spheres localized on the solutions to a set of constraints that relate the punctures over the Riemann spheres to the kinematic invariants of the process, known as the scattering equations. We will introduce the CHY formalism for pure Yang-Mills amplitudes, as well as one of its supersymmetric generalizations, the so-called maximally supersymmetric or N = 4 super Yang-Mills theory (SYM). We will introduce the notion of basis amplitudes for Yang-Mills and Quantum Chromodynamics (QCD), which are based on the idea of color decomposition, the separation of color and kinematic degrees of freedom, and see how one can obtain CHY representations for QCD amplitudes. We will use one of these representations, given in terms of basis amplitudes, to derive soft theorems for the CHY integrand of QCD, which is a first step into obtaining constraints on its mathematical structure.