A Diagrammatic Study of Gauge and Gravity Amplitude Relations

We review modern methods of scattering amplitude computations, beginning with color-decomposition that decomposes the amplitude into a color-dependentand a color-independent part of partial color-ordered amplitudes. We then show how a full amplitude can be described through the color-factor, propagator structure, and kinematic numerator of cubic color-ordered diagrams. The color-factors obey a Lie algebra and as such satisfy the Jacobi identity. We are able to impose the color-kinematics duality that states that the kinematic numerators also obey this identity. Because of this it is possible to write down sets of Jacobi equations for the numerator through their diagrammatic expression. These can be solved for a set of master numerators through which all other numerators can be expressed linearly. We find such solutions for the tree-level, one-loop, and two-loop diagrams for any numberof particles