The Compton Amplitude from the Lattice Feynman-Hellmann Method

Abstract

In both experiment and theory, Compton scattering provides a unique window into the internal structure of hadrons. At high energies the Compton amplitude is parameterised by parton distributions, which describe the properties of hadrons in terms of their fundamental constituents, quarks, anti-quarks and gluons - a vital bridge between the quark-gluon picture of quantum chromodynamics (QCD) and the hadronic picture that characterises experiment. Despite their importance, the Compton amplitude and parton distributions have both proven difficult to determine from fi rst principles QCD. In this thesis, we perform the fi rst calculations of the unpolarised off-forward and polarised forward Compton amplitudes in lattice QCD. By extending the Feynman-Hellmann theorem to second-order, we are able to calculate these amplitudes from lattice two-point functions computed in the presence of a background field, and thereby overcome difficulties associated with direct calculations. Since we determine the Compton amplitude, we not only have the potential to determine parton distributions, but also a wealth of complementary properties such as scaling behaviour, higher-twist effects, and the subtraction function. For the present investigation, we focus on determining the Mellin moments of these amplitudes. In both the unpolarised off-forward and polarised forward cases, we nd that our leading moments agree reasonably well with both phenomenological expectations and determinations in other lattice methods. However, in attempting to constrain higher moments and reconstruct the parton distributions, we encounter a range of difficulties. We discuss key lattice systematics and identify strategies to overcome these in future work. Following this discussion of lattice systematics, we devote the last chapter to an investigation of short-distance artefacts affecting our calculations. Focusing on the Compton amplitude subtraction function, we show that such short-distance artefacts are signi ficant. However, we also demonstrate that these artefacts can be controlled using a range of tools including varying the discretisation and an analytic expansion, thus paving the way for much improved calculations in future work. The ultimate aim of our method is a fi rst principles calculation of the Compton amplitude with good control of all systematics. Such a calculation would have a far-reaching impact on our understanding of nucleon structure, in areas as varied as the proton spin puzzle, the proton-neutron mass difference, the proton radius puzzle and the strong coupling in the con fined regime - not to mention the immense signifi cance of the parton distributions themselves. This thesis takes us a step closer to that goal, extending the Feynman- Hellmann method to new kinematics and spin-dependent amplitudes, and starting the work to address key systematics.