Doubleβ-decay nuclear matrix elements and lepton conservation

The nuclear matrix elements involved in the double $\beta$-decay of ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$, ${}_{}{}^{130}{}_{}{}^{}\mathrm{Te}$, and ${}_{}{}^{128}{}_{}{}^{}\mathrm{Te}$ were calculated using realistic nuclear interactions and shell model nuclear wave functions. The double doorway state is not appreciably mixed in the ground state of the final nuclei. So the ground state transitions contain a small fraction of the sum rule. A lepton nonconservation parameter $\eta \lesssim {10}^{-4\mathrm{}}$ was deduced.