Effective Field Theory and Approximate Symmetries for Low-Energy Few-Body Systems

Abstract

Effective field theory (EFT) is a powerful tool for studying physical systems characterized by separated length scales. An EFT captures essential features of a physical system at a certain scale by including interactions informed by symmetries and expanded in ratios between length scales. In this way, an EFT provides a relatively simple way to obtain systematically improvable predictions of observables at a certain scale. Pionless EFT, one of the low-energy EFTs of quantum chromodynamics (QCD), has proven its success in describing few-nucleon systems. As a member of a broader class of short-range EFTs consisting of contact interactions, Pionless EFT has a well-understood renormalization, displays a high degree of universality, and can be used to study few-nucleon systems semi-analytically. In this thesis, we present our Pionless EFT studies of cold neutron-deuteron capture into the triton and a photon ($nd\to \triton \gamma$) and dark matter (DM) scattering off light nuclei; we also present a short-range EFT study of the four-boson system. We incorporate approximate symmetries in our studies to help understand these processes. For cold $nd$ capture, we calculate the cold $nd$ capture cross section ($\sigma_{nd}$) up to and including next-to-next-to-leading order (NNLO) in Pionless EFT and use the Wigner-SU(4) symmetry to understand the suppression on the contribution from the single-nucleon magnetic currents, as observed in previous calculations using potential models or other EFTs. We also identify a three-nucleon magnetic moment counterterm needed to renormalize both $\sigma_{nd}$ and the triton magnetic moment at NNLO. For DM-light-nuclei scattering, we compute the DM-nuclei cross section for $A\leq 3$ up to and including next-to-leading order in Pionless EFT and use the large-$N_c$ (number of QCD colors) expansion to constrain the contribution from different one- and two-nucleon-DM interactions; this study helps understand how DM may interaction with nucleons with future experiments using light nuclei as targets. For the four-boson system where discrete scaling symmetry plays a crucial role, we investigate the renormalization of four-body binding energies for cold $^4$He atoms and their behavior near the unitary limit; this calculation is also a precursor to four-nucleon calculations.