Doubleβ-decay nuclear matrix elements for theA=48andA=58systems

The nuclear matrix elements entering the double $\beta$ decays of the ${}_{}{}^{48}{}_{}{}^{}\mathrm{Ca}$-${}_{}{}^{48}{}_{}{}^{}\mathrm{Ti}$ and ${}_{}{}^{58}{}_{}{}^{}\mathrm{Ni}$-${}_{}{}^{58}{}_{}{}^{}\mathrm{Fe}$ systems have been calculated using a realistic two nucleon interaction and realistic shell model spaces. Effective transition operators corresponding to a variety of gauge theory models have been considered. The stability of such matrix elements against variations of the nuclear parameters is examined. Appropriate lepton violating parameters are extracted from the $A=48\mathrm{}$ data and predictions are made for the lifetimes of the positron decays of the $A=58\mathrm{}$ system.