Hadron Structure Using Feynman-Hellmann Theorem

Abstract

Hadrons, such as protons and neutrons, are states that are formed through interactions of quarks and gluons, the fundamental building blocks of Quantum Chromodynamics (QCD). The role of non–perturbative effects in the emergent behaviour of QCD is a key ingredient in our understanding hadrons and hence the atoms formed thereof. These dynamics have important consequences for matter in the universe from the atomic scale to neutron stars and beyond. We use an ab initio non–perturbative numerical path integral based approach to QCD, known as lattice QCD. Advancing computing resources have made possible rapid advances in hadronic studies in lattice QCD, but many challenges still remain. Two areas of vital importance to our understanding of QCD and future experiments are gluonic observables and structure functions. Gluonic observables are difficult to calculate on the lattice due to sensitivity to short distance gauge noise. Naïve structure function calculations suffer from rapidly increasing computational cost as the lattice grows to a size where discretisation systematics are under control, as well as problems matching onto Minkowski matrix elements. A modification to the QCD action changes the energy eigenstates of hadrons. The shift in these eigenstates can be related to matrix elements with interactions introduced in the shifted action via the Feynman–Hellmann Theorem (FHT). We show how the FHT can be extended to second order to calculate two current hadronic matrix elements using only two–point function techniques. A detailed analysis on how to improve uncertainty and reduce computational requirements of any FHT calculation is given. Using the FHT the full Compton amplitude is calculated, which allow us to explore assumptions made in experimental parton studies in Deep Inelastic Scattering (DIS). The subtraction function, given in terms of the Compton amplitude is not experimentally extractable and is examined from first–principles for the first time. Gluonic matrix elements are traditionally difficult to calculate on the lattice. By using Wilson flow to reduce short distance effects, forward matrix elements are determinable with reduced uncertainty. By classification of the Lorenz structure of off–forward gluonic matrix elements, the extraction of non–forward matrix elements were also made possible, providing further insight into the highly non–perturbative binding of hadrons.