  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.
