Ordinary muon capture as a probe of virtual transitions of $\beta\beta$ decay

A reliable theoretical description of double-beta-decay processes needs a possibility to test the involved virtual transitions against experimental data. Unfortunately, only the lowest virtual transition can be probed by the traditional electron-capture or $\beta^-$-decay experiments. In this article we propose that calculated amplitudes for many virtual transitions can be probed by experiments measuring rates of ordinary muon capture (OMC) to the relevant intermediate states. The first results from such experiments are expected to appear soon. As an example we discuss the $\beta\beta$ decays of $^{76}$Ge and $^{106}$Cd and the corresponding OMC for the $^{76}$Se and $^{106}$Cd nuclei in the framework of the proton-neutron QRPA with realistic interactions. It is found that the OMC observables, just like the $2\nu\beta\beta$-decay amplitudes, strongly depend on the strength of the particle-particle part of the proton-neutron interaction.