Searches for neutrinoless double-$\beta$ decay rates are crucial in addressing questions within fundamental symmetries and neutrino physics. The rates of these decays depend not only on unknown parameters associated with neutrinos, but also on nuclear properties. In order to reliably extract information about the neutrino, one needs an accurate treatment of the complex many-body dynamics of the nucleus. Neutrinoless double-$\beta$ decays take place at momentum transfers on the order of $100\mathrm{MeV}/c$ and require both nuclear electroweak vector and axial current matrix elements. Muon capture, a process in the same momentum transfer regime, has readily available experimental data to validate these currents. In this Letter, we present results of ab initio calculations of partial muon capture rates for ${}^3\mathrm{He}$ and ${}^6\mathrm{Li}$ nuclei using variational and Green's function Monte Carlo computational methods. We estimate the impact of the three-nucleon interactions, the cutoffs used to regularize two-nucleon $(2N)$ interactions, and the energy range of $2N$ scattering data used to fit these interactions.

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• Electron & muon capture
• Electroweak interactions in nuclear physics
• Ab initio calculations
• Nuclear many-body theory
• Nuclear structure & decays
• Quantum Monte Carlo