Determination of Strength of Isoscalar Pairing Interaction by a Mathematical Identity in QRPA

I propose a new method to determine the strength of the isoscalar proton-neutron pairing interaction by a mathematical identity derived in the quasiparticle random-phase approximation. This method is applied for a few nuclei possibly having the neutrinoless double-$\beta$ decay. Reduced half-life, the theoretical quantity necessary for determining the effective neutrino mass, is calculated for $^{48}$Ca.