To describe the double-charge-exchange (DCE) processes, we have designed recently the $(pn,2p2n)$-QTDA model which fully includes the pairing correlations and four quasiparticle excitations. It has been applied in $2 u$ double beta decays (DBDs), and the double charge-exchange resonances (DCERs). Here we extend it to $ 0 u $ DBD and discuss the relationship between the nuclear matrix elements (NMEs), and the DCE reaction matrix elements (RMEs) with the same spin-isospin structure. We do it for all final $0^+$ states, even in the region of DCERs, where the DBD is energetically forbidden. As an example, we evaluate the DBD $^{76}$Ge $rightarrow ^{76}$Se, both for $2 u$ and $0 u$ modes, as well as the associated DCE sum rules, excitation energies within the $Q$-value window for DBD, and the $Q$-value itself. We find that the $0 u$ NMEs are correlated with the RMEs, both at low energy, and in the DCER region where most of the transition strength is concentrated. These findings occur in other nuclei as well and suggest that measurements of $0^+$ DCERs could provide useful information regarding the $ 0 u $ DBD. An analogous comparison and conclusion cannot be made for the $2^+$ states, since the $0 u$ NMEs and RMEs transition operators are not similar to each other in this case.
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