We discuss two conditions needed for correct computation of $2\nu \beta\beta$ nuclear matrix-elements within the realistic shell-model framework. An algorithm in which intermediate states are treated based on Whitehead's moment method is inspected, by taking examples of the double GT$^+$ transitions $\mbox{ $^{ 36}$Ar}\rightarrow\mbox{ $^{ 36}$S}$, $\mbox{ $^{ 54}$Fe}\rightarrow\mbox{ $^{ 54}$Cr}$ and $\mbox{ $^{ 58}$Ni} \rightarrow\mbox{ $^{ 58}$Fe}$. This algorithm yields rapid convergence on the $2\nu\beta\beta$ matrix-elements, even when neither relevant GT$^+$ nor GT$^-$ strength distribution is convergent. A significant role of the shell structure is pointed out, which makes the $2\nu\beta \beta$ matrix-elements highly dominated by the low-lying intermediate states. Experimental information of the low-lying GT$^\pm$ strengths is strongly desired. Half-lives of $T^{ 2\nu}_{ 1/2}({ \rm EC}/{ \rm EC}; \mbox{ $^{ 36}$Ar}\rightarrow\mbox{ $^{ 36}$S})=1.7\times 10^{ 29}\mbox{ yr}$, $T^{ 2\nu}_{ 1/2}({ \rm EC}/{ \rm EC};\mbox{ $^{ 54}$Fe}\rightarrow \mbox{ $^{ 54}$Cr})=1.5\times 10^{ 27}\mbox{ yr}$,$T^{ 2\nu}_{ 1/2}({ \rm EC} /{ \rm EC};\mbox{ $^{ 58}$Ni}\rightarrow\mbox{ $^{ 58}$Fe})=6.1\times 10^{ 24}\mbox{ yr}$and $T^{ 2\nu}_{ 1/2}(\beta^+/{ \rm EC};\mbox{ $^{ 58}$Ni} \rightarrow\mbox{ $^{ 58}$Fe})=8.6\times 10^{ 25}\mbox{ yr}$ are obtained from the present realistic shell-model calculation of the nuclear matrix-elements.
Categories: nucl-th