## 1 Introduction  
The theoretical study of the hypothesized rare neutrinoless double beta (\(0\
u\beta\beta\)) decay is challenging, yet it is among the most promising avenues of physics research beyond the standard model [1, 2, 3, 4, 5]. The complexity in the study of \(0\
u\beta\beta\) decay stems from the involvement of nuclear-structure effects/correlations from low to high momentum-exchange scales (\(q\leq 100-200\) MeV) and nuclear states of high energy and/or multipolarity (\(J^{\pi}\)). Experimental nuclear-structure data at medium and high momentum scales are seldom available and are almost entirely uncharted territory, making it difficult for nuclear models and, hence, the computed \(0\
u\beta\beta\) nuclear matrix elements (NMEs) to be improved upon being tuned to such data. Given that \(0\
u\beta\beta\) decay has not been measured, accurate nuclear modeling of this process for various \(0\
u\beta\beta\)-decay candidates is essential for determining the sensitivity of experiments designed to detect this rare decay [6]. There are significant discrepancies in the \(0\
u\beta\beta\)-decay NMEs computed in various nuclear-model frameworks [1], and imperfect nuclear-structure calculations demand the use of an effective value of the axial vector coupling (\(g_{A}^{eff}\)) [5, 6, 7, 8]. Discrepancies in \(0\
u\beta\beta\)-decay NMEs across nuclear models and uncertainty in the value of \(g_{A}^{eff}\) propagate in the 2 nd and 4 th powers, respectively, to the computed/predicted half-lives. Ordinary muon capture (OMC) is a seemingly miraculous process in that it is the only known practical way to systematically investigate the nuclear structure experimentally at momentum scales relevant to the physics of \(0\
u\beta\beta\) decay [9, 10, 11, 12]. OMC can also populate all the nuclear states that are intermediate states of the odd-odd nucleus, via which the \(0\
u\beta\beta\) decay proceeds [9; 10; 11; 12]. This means that OMC can and is used to access decay amplitudes of one leg, involving either the daughter or the mother nucleus of the \"two-step\" rare transition, depending on whether \(0\
u\beta\beta\) is of \(\beta^{-}\) or \(\beta^{+}\)/EC type, respectively [9; 10; 11; 12; 13]. Involvement of common decay amplitudes in this way for the computed OMC NMEs/rates and \(0\
u\beta\beta\)-decay NMEs leads us to look for connections between the two. Towards both ends of computing more accurate \(0\
u\beta\beta\)-decay NMEs and determination of \(s_{A}^{eff}\), the OMC process is a gift from nature as it can help address both of these goals. The study of OMC is the best-known way to test the fitness of nuclear models and improve their accuracy, by closely tuning them to experimental data for computing physically relevant OMC NMEs. Tuning the nuclear models this way makes them optimized to compute \(0\
u\beta\beta\)-decay amplitudes and ultimately NMEs due to similar nuclear and weak-interaction contributions involved in the two processes [10; 11; 12; 14]. As an example, OMC can give us access to the value of the particle-particle interaction parameter (\(g_{pp}\)) in the pnQRPA (proton-neutron quasiparticle random-phase approximation) framework [10; 11; 12], and \(g_{A}^{eff}\) and/or the effective value of the pseudoscalar coupling (\(g_{P}\)) [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29] at momentum scales relevant for \(0\
u\beta\beta\) decay. Recent calculations of OMC rates were performed in [30]. Only in the recent past have major experimental efforts been made to leverage OMC to illuminate further the mystery of \(0\
u\beta\beta\) decay, by measuring (partial) OMC rates, as performed in present state-of-the-art experiments such as the MONUMENT experiment [31]. Such experiments will make available invaluable experimental constraints for grounding the theoretical modeling of OMC processes, offering a tangible map for the improvement of nuclear models, and leading the way to more accurate computed OMC/\(0\
u\beta\beta\)-decay NMEs.