Hansen, MaxwellAlotaibi, Athari2025-11-042025https://hdl.handle.net/20.500.14154/76818The strong force, responsible for binding quarks and gluons inside hadrons, is described by the mathematical framework of quantum chromodynamics (QCD). One approach to studying the non-perturbative region of this theory is lattice QCD. This numerical method extracts the hadron spectrum and other physical observables from first principles by formulating QCD on a discretized Euclidean spacetime lattice with finite volume. Lattice QCD combined with the finite-volume formalism, enables the extraction of infinite-volume scattering observables from the discrete energy spectrum obtained in lattice simulations. This formalism is derived from relativistic-field-theoretic analysis of the poles of the finite-volume scattering amplitude and realized via a quantization condition, whose solutions give the finite spectrum. Over the last decades, the formalism has matured from treating two-particle to three-particle systems with sub-resonant channels, including odd-legged vertices that allow for two-to-three scattering. This thesis derives and implements novel theoretical results concerning two key aspects of lattice QCD scattering calculations. First, we implement the formalism for the three-pion system in all allowed non-maximal isospin channels to numerically compute the finite-volume spectrum using Python called ampyL. This is performed across multiple moving frames for various irreducible representations of the finite-volume symmetry group. The system includes two-pion resonant subchannels associated with the ρ and σ resonances, with vanishing three-body interactions. These results serve as a benchmark and database for future lattice calculations aimed at extracting three-pion resonances. Second, we implement the coupled quantization condition, i.e., including 2-to-3 scattering, for identical scalar particles in a rest frame to extract the energy spectrum. Then, we focus on the avoided-level crossing that occurs when a two- and three-particle level intersect in the spectrum and show that the spectrum in this region follows the form of a rotated hyperbola, leading to a simple analytic expression that can be fitted to lattice data in weakly interacting systems. This offers a practical alternative to using the full quantization condition.166enLattice QCDFinite Volume PhysicsPion ScatteringThesis