Experimental studies of nuclear matrix elements (NMEs) for neutrinoless double-$\beta$ decays (DBDs) and astro-neutrino $\left(\nu \right)$ inverse $\beta$ decays (IBDs) are crucial for $\nu$ studies beyond the standard model and the astro-$\nu$ studies since accurate theoretical calculations of the NMEs are hard due to the high sensitivity of the NMEs to the nuclear models and the nuclear parameters used for the models. Some of the important NMEs of electromagnetic transition operators associated with DBD and IBD, including the effective weak couplings, are found to be experimentally obtained by measuring the corresponding electromagnetic gamma (EM:$\gamma )$ transitions from the isobaric analog states (IASs) of the DBD and IBD nuclei. Then the experimental NMEs and the couplings are used for evaluating the DBD and IBD NMEs and for checking the model calculations. The EM-NMEs, the cross sections, and the event rates for the IAS-$\gamma$ transitions are estimated for DBD and IBD nuclei to show the feasibility of the experiments.

• Beta decay
• Electromagnetic transitions
• Electroweak interaction
• Nuclear astrophysics
• Nuclear structure & decays