Toward Reliable $0\nu\beta\beta$-decay Nuclear Matrix Elements: Exploring the potential of measuring $\gamma\gamma$-transitions

In this work, a physics process known since quite long ago, double-gamma decay ($\gamma\gamma$), has been revisited from a new perspective: providing valuable insights into neutrinoless double-beta decay ($0\nu\beta\beta$) nuclear matrix elements. At the same time that an eager experimental search of $0\nu\beta\beta$ decay is underway, the nuclear and particle physics communities have made huge progress during the last years. The main goal of this thesis has been to investigate one of those approaches which is the computation of nuclear observables related to $0\nu\beta\beta$ decay as a way to help in determining and reducing the theoretical uncertainties in $0\nu\beta\beta$-decay NMEs. This way, we have proposed that the measurement of the double magnetic dipole $\gamma\gamma$ decay from the double isobaric analog state in the $\beta\beta$-decay final nucleus could establish the value and reduce the uncertainty in $0\nu\beta\beta$-decay NMEs, because the NMEs of the two processes are very well correlated. We have explored the validity of this approach to predict and quantify NMEs uncertainties in the case where data is available for the nuclear observable related to $0\nu\beta\beta$ decay, that is for the Standard Model allowed $2\nu\beta\beta$ decay. A further objective has been to start with the theoretical characterization of the first steps toward the measurement of the proposed $\gamma\gamma$ double magnetic dipole decay from the double isobaric analogue state. In particular, we have studied the main decaying modes that can compete with this process: single gamma decay and proton emission, vi and we have calculated the corresponding branching ratios. In addition, we have also given the first steps in the study of the relation between $\gamma\gamma$ and $0\nu\beta\beta$-decay NMEs in the ab initio valence space in-medium similarity renormalization group.