Physical features of strength of isoscalar pairing interaction determined by relation between double charge change and double pair transfer

A new method has been proposed to determine the strength of the isoscalar proton-neutron pairing interaction applicable to many nuclei. The principle is the equivalence between the double charge change and the double transfer of like-particle pair, and a constraint is derived to the effective interactions used in approximations. This method was applied to the quasiparticle random-phase approximation for determining that interaction strength. In this paper, detail of this method is explained thoroughly, and applications are made to nuclei of several instances of the double-$\beta$ decays. The systematics of the strengths determined for those nuclei is understood in terms of a midshell effect. The effect of the new interaction strength is examined in two examples of the Gamow-Teller strength function with comparisons with the experimental data. The nuclear matrix elements of the neutrinoless double-$\beta$ decay are also calculated.