Advances to Three-Particle Quantization Conditions: Hadron Scattering Amplitudes from Lattice QCD

Thesis (Ph.D.)--University of Washington, 2021The strong nuclear force is responsible for the vast majority of particles in nature; hundreds of hadrons with distinct masses and quantum numbers have been detected, and experimentalists are constantly searching for new hadron resonances and analyzing resonance decays to test the predictions of the Standard Model (e.g.~by measuring the value of a $CP$-violating phase).While the formalism of quantum chromodynamics (QCD) appears to accurately describe the strong interaction, using it to extract theoretical predictions of hadronic properties has proven exceedingly difficult due to the nonperturbative nature of low-energy QCD. Lattice QCD (LQCD) is the state-of-the-art approach for computing physical observables using first-principles QCD, utilizing stochastic techniques to estimate the path-integral expressions for correlation functions in a latticized Euclidean spacetime. With the computational and algorithmic advances over the past few decades, LQCD can now be used to calculate many observables (e.g.~hadron masses) to percent-level precision. However, LQCD simulations cannot directly calculate scattering amplitudes, due to both the necessary use of a finite volume and the need to analytically continue from Euclidean to Minkowski space, and thus a separate formalism is required. To bridge the gap between LQCD results and hadron spectroscopy, one needs some form of quantization condition (QC): a mathematical relation used to extract physical scattering amplitudes of light hadrons from finite-volume energy spectra calculated with LQCD. In this thesis, we present several theoretical and practical developments in the field of three-particle QCs (QC3s). We derive three significant advances to the theoretical formalism of QC3s within a generic relativistic effective field theory (RFT) framework. First, we perform an alternative derivation of the original QC3 for three identical scalars by using time-ordered perturbation theory (TOPT), which we find significantly simplifies the argument and thus renders the TOPT approach more amenable to generalization. We then demonstrate this point by using TOPT to derive a QC3 for the more complicated case of three nondegenerate scalars. Lastly, we use results from the TOPT approach to demonstrate the equivalence between the RFT QC3 for identical scalars and that of the finite-volume unitarity (FVU) approach. We also present three numerical applications of QC3s to systems of three identical scalars.We detail the first QC3 study to include levels in nontrivial irreducible representations (irreps) of the cubic group and with nonzero $d$-wave two-particle interactions. Next, we perform an exploratory investigation of bound states and resonances by solving two- and three-particle QCs in various scenarios. Finally, we present the first ever extraction of $3\pi^+$ scattering information from LQCD data