  Muon capture on nuclei is one of the most promising probes of the nuclear
electroweak current driving the yet-hypothetical neutrinoless double-beta
($0\nu\beta\beta$) decay. Both processes involve vector and axial-vector
currents at finite momentum transfer, $q\sim 100$ MeV, as well as the induced
pseudoscalar and weak-magnetism currents. Comparing measured muon-capture rates
with reliable $ab~initio$ nuclear-theory predictions could help us validate
these currents. To this end, we compute partial muon-capture rates for
$^{6}$Li, $^{12}$C and $^{16}$O, feeding the ground and excited states in
$^{6}$He, $^{12}$B and $^{16}$N, using $ab~initio$ no-core shell model with
two- and three-nucleon chiral interactions. We remove the spurious
center-of-mass motion by introducing translationally invariant operators and
approximate the effect of hadronic two-body currents by Fermi-gas model. We
solve the bound-muon wave function from the Dirac wave equations in the Coulomb
field created by a finite nucleus. We find that the computed rates to the
low-lying states in the final nuclei are in good agreement with the measured
counterparts. We highlight sensitivity of some of the transitions to the
sub-leading three-nucleon interaction terms. We also compare summed rates to
several tens of final states with the measured total capture rates and note
that we slightly underestimate the total rate with this simple approach due to
limited range of excitation energies.
