Consistency Examinations of Calculations of Nuclear Matrix Elements of Double-$β$ Decay by QRPA

The neutrinoless double-$β$ decay is a hypothetical rare nuclear decay, which can be used for determining the neutrino-mass scale. The scheme to use this decay for determining the neutrino-mass scale is one of few limited methods possible to determine that. Nuclear matrix element of this decay is an important input to this method, and this matrix element cannot be determined by experiment. I examine the validity of the transition density used for calculating the nuclear matrix element by comparing the experimental data and my calculated result of the charge-change strength functions of $^{48}$Ca and $^{48}$Ti. The nuclear wave functions are obtained by the quasiparticle random-phase approximation. A new idea is proposed on the transition operator for this strength function, and the data of those nuclei are reproduced well consistently. Reduced half-life of a few nuclei to the neutrinoless double-$β$ decay are shown.