Electroweak Properties of Halo Nuclei in Effective Field Theory

The exploration of atomic nuclei across the nuclear landscape poses a great challenge in nuclear theory. In this thesis, we investigate nuclei on the nuclear chart for which the effective number of degrees of freedom is significantly smaller than the number of nucleons. This phenomenon is called clustering and becomes extreme for so-called halo nuclei. These exotic nuclei occur along the neutron and proton driplines far from the valley of stability. They exhibit a scale separation apparent through the small separation energy of the loosely bound valence nucleons in contrast to the high binding and excitation energies of the core. By exploiting this scale separation, we construct an effective field theory (EFT) called Halo EFT which allows to describe these systems in a controlled and systematically improvable manner. In the first part, we investigate the one-neutron halo structures within ¹⁵C. They appear in the ground 1/2⁺ and in the first excited 5/2⁺ state. The ground state is predominantly bound in an S-wave while the 5/2⁺ excited state is predominantly bound in a D-wave. Within Halo EFT, using standard Cartesian coordinates, we discuss static electromagnetic properties as well as electromagnetic transitions in halo nuclei and apply our results to ¹⁵C. Since our results are universal, we are able to compare them to ab initio results. This possibility enables us to determine unknown low-energy constants, in turn increasing the predictive power of our Halo EFT. The second system is the one-neutron halo nucleus ³¹Ne which is bound in a P-wave. In this case, we construct a Halo EFT using a spherical basis, an approach ideally suited for the inclusion of halo states beyond the S-wave. Thereby, we investigate the electromagnetic E1 breakup reaction into the continuum consisting of the neutron and ³⁰Ne core. Additionally, we provide results for static properties and discuss the deformation of ³¹Ne due to the non-vanishing quadrupole moment. The third part is a pilot study of the weak decay of the valence neutron of the halo nucleus ¹¹Be into the continuum within Halo EFT. This process, denoted ¹¹Be → ¹⁰Be + p + e⁻ + ν̅ₑ, is called beta-delayed proton emission. The experimental determination of the branching ratio for this decay remains an unsolved problem due to inconsistent measurements in different experiments. We calculate the rate of this rare decay with a robust uncertainty estimate. We also discuss the impact of a recently discovered resonance in ¹¹B on the branching ratio and compare it to different experimental results. In the fourth and final part of this thesis, we investigate the universal behavior of weakly bound charged systems in three- and one-dimensional space. The focus lies on the study of one-proton halo nuclei bound in an S-wave. In particular, the impact of the repulsive long-ranged Coulomb force is analyzed. It introduces an additional length scale D (or momentum scale kᴄ). We classify universal regimes characterized by the different hierarchies of the Coulomb-modified scattering length aᴄ and D