$\textit{Ab initio}$ calculation of the calorimetric electron capture spectrum of $^{163}$Holmium: Intra-atomic decay into bound-states

The determination of the electron neutrino mass by electron capture in $^{163}$Ho relies on a precise understanding of the deexcitation of a core hole after an electron capture event. We here present an \textit{ab intio} calculation of the electron capture spectrum in $^{163}$Ho, including all intra-atomic decay channels into bound-states. We use theoretical methods developed for the calculation of core level spectroscopy on correlated electron compounds. Our comparison critically tests the reality of these theories. We find that relativistic interactions beyond the Dirac equation, i.e. quantum-electro dynamics, only lead to minor shifts of the spectral peaks. The electronic relaxation after an electron capture event due to the changed nuclear potential leads to a mixing of different edges, but due to conservation of angular momentum of each scattered electron, no additional structures emerge. Many-body Coulomb interactions lead to the formation of multiplets and to additional peaks with multiple core-holes due to Auger decay. Multiplets crucially change the appearance of the resonances on a Rydberg energy scale. The additional structures due to Auger decay are, although clearly visible, relatively weak compared to the one core hole states and accidentally far away from the end-point region of the spectrum. As the end-point of the spectrum is effected most by the neutrino mass these additional states do not influence the statistics for determining the neutrino mass directly. The multiplet broadening and Auger shake-up of the main core-level edges do change the apparent line-width and accompanying lifetime of these edges, thereby invalidating experimentally obtained lifetimes at the resonance for regions far away from the resonance.