Young, RossZanotti, JamesHowson, Tomas Liam2024-07-082024-07-082024https://hdl.handle.net/2440/141555The hadrons are the group of subatomic particles that form the building blocks for visible matter in our universe. Composed of quarks and gluons, understanding the structure of these hadrons is crucial for a complete understanding of the composition of matter. This work investigates an aspect of the hadron structure known as the momentum fractions, which describe the amount of hadron’s total momentum carried each by the quarks and gluons. With upcoming experiments capable of probing this structure, such as the Electron-Ion Collider at Brookhaven National Laboratory, a strong theoretical underpinning of the hadron structure is crucial in the formation of our understanding. The theory describing the internal dynamics of quarks and gluons in hadrons is Quantum Chromodynamics (QCD), a gauge field theory (GFT). Due to the highly self-interacting nature of QCD at low energies, perturbative methods to calculate observables break down. This work uses the framework of lattice QCD, a formulation of QCD that allows for non-perturbative calculations using path integral approaches defined on a discretised spacetime. Lattice QCD is equipped with many conventional techniques for the calculation of operators acting on particle states. In this thesis we examine an application of the Feynman-Hellmann method, an alternative to the conventional techniques. Of notable interest is the calculation of disconnected contributions to the matrix elements of operators, in particular the gluon and sea quark momentum fractions, where signals typically suffer from strong statistical noise, and require stochastic techniques to reliable produce results. This work shows that the Feynman-Hellmann method produces compatible results with conventional techniques for calculating the connected contribution to operators, and examines the viability of the Feynman-Hellmann in calculating disconnected contributions to operators. We implement the first use of the Feynman-Hellmann method to calculate the disconnected quark momentum fractions from the energy-momentum tensor. To compare lattice calculated momentum fractions with continuous space quantities, we must consider the renormalisation of such values. Here we implement the Feynman-Hellmann method to create a fully non-perturbative renormalisation scheme, capable of capturing the mixing between the quark and gluon sectors. This work shows the first use of the Feynman-Hellmann method to produce all renormalisation factors required to renormalise the momentum fractions non-perturbatively, including both quark and gluon components, and the mixing between. The method as demonstrated produces statistically significant signals, that are consistent with the expected sum rules, for a modest computational cost. The renormalised results obtained show the gluon as comprising a significant portion of the momentum content of the hadrons, at a similar magnitude to the quark component. This result is consistent between nucleons and pions, and holds in both the quenched approximation and in the presence of dynamically generated quarks. Through this work, we demonstrate the Feynman-Hellmann method as producing results that are self-consistent, and compatible with other technique’s results for the momentum fractions. This shows the Feynman- Hellmann method is a powerful technique, that could prove useful as part of future analysis into disconnected contributions to operator matrix elements.enLattice QCDHadron StructureRenormalisationThesis