  In this work we study ordinary muon capture (OMC) on $^{24}$Mg from a first
principles perspective. Starting from a particular two- and three-nucleon
interaction derived from chiral effective field theory, we use the
valence-space in-medium similarity renormalization group (VS-IMSRG) framework
to construct effective Hamiltonians and muon-capture operators which
nonperturbatively account for many-body physics outside the valence space. The
obtained nuclear matrix elements are compared against those from the
phenomenological shell model. The impact of including the correlations from the
nuclear shell model (NSM) as well as including the induced two-body part is
studied in detail. Furthermore, the effects of realistic bound-muon wave
function on the operators is studied. Finally, predictions for capture rates to
the lowest excited states in $^{24}$Na are given and compared with available
data. It is found that the spectroscopic properties of $^{24}$Mg and its OMC
daughter $^{24}$Na are fairly well described by both the NSM and VS-IMSRG, and
that the effect of the hadronic two-body currents significantly reduces the OMC
rates. Both models have some difficulties in matching the measured OMC rates,
especially for the $2^+$ final states. This calls for further studies in other
light nuclei with available OMC data.
