QCD Representations of Quark and Gluon Spin and Orbital Angular Momentum

This thesis presents the derivation of Lorentz Invariance and Equation of Motion Relations between theoretical objects that involve the partonic transeverse momentum called Generalized Transverse Momentum Distributions (GTMDs) and collinear functions known as Generalized Parton Distributions (GPDs) that describe quark and gluon orbital angular momentum. Although the GTMDs in principle define the observables for partonic orbital motion, experiments that can unambiguously detect them appear remote at present. The relations presented here provide a solution by showing how, for instance, the orbital angular momentum density is connected to directly measurable twist-three GPDs.

While experimental measurement is the only certain way to access the unknown functions that describe quarks inside a proton such as the Parton Distribution Functions (PDFs) and GPDs that characterize the nucleon, a great deal of effort has gone into evaluating these functions using theoretical techniques such as model calculations and parameterizations. Lattice QCD provides the only first principle calculations of the Mellin moments of PDFs and GPDs. The work prsented here shows how, using only a few moments, a large portion of the Fourier transform with respect to Bjorken $x$ of the PDFs and GPDs can be mapped out. In the case of GPDs, lattice calculations provide a value for the Compton form factors that allow us to move away from the $x = \xi$ ridge which, at present, is the only area of the phase space that has been explored experimentally. After studying how well known parameterizations of the PDFs and GPDs reproduce test functions, we demonstrate how PDFs and GPDs can be reconstructed using the moments calculated by lattice QCD.