Strength of the isoscalar pairing interaction determined by a relation between double-charge change and double-pair transfer for double-$\beta$ decay

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 pairs, and a constraint is derived for the effective interactions used in approximations. This method was applied to the quasiparticle random-phase approximation for determining that interaction strength. In this paper, details of this method are explained thoroughly, and applications to nuclei of several instances of the double-$\beta$ decays are shown. 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.

Figure 1

GT component $M_{\mathrm{GT}}^{\left(0\nu \right)}$ of $0\nu \beta \beta$ nuclear matrix element for ${}^{150}\mathrm{Nd}–{}^{150}\mathrm{Sm}$ calculated by the $pn\text{QRPA}$ as a function of $G_{pn}^{\mathrm{IS}}$ (solid line) and that calculated by the lpQRPA (dashed line). The latter is independent of the isoscalar pairing interaction; thus, it is drawn as a constant line.

Figure 2

The same as Fig. 1 but for ${}^{136}\mathrm{Xe}–{}^{136}\mathrm{Ba}$.

Figure 3

The same as Fig. 1 but for ${}^{48}\mathrm{Ca}–{}^{48}\mathrm{Ti}$.

Figure 4

$-G_{pn}^{\mathrm{IS}}$ as a function of $A$. The values are noted in Table 3.

Figure 5

$-G_{pn}^{\mathrm{IS}}$ as a function of ${\mathcal{N}}_{\mathrm{lp}I}{\mathcal{N}}_{\mathrm{lp}F}/A$ with $A$ inserted. For ${\mathcal{N}}_{\mathrm{lp}I}$ and ${\mathcal{N}}_{\mathrm{lp}F}$, see Sec. 2.

Figure 6

${\mathcal{N}}_{\mathrm{lp}I}{\mathcal{N}}_{\mathrm{lp}F}$ as a function of $A$.

Figure 7

Charge-change strength function by the GT and isovector spin monopole (IVSM) operator for transition from ${}^{48}\mathrm{Ca}$ to ${}^{48}\mathrm{Ti}$. $E$ is the excitation energy of the final nucleus. The calculated result is shown by line, and the experimental data [49] are shown by isolated symbols: (a) The calculation was performed with $G_{pn}^{\mathrm{IS}}=-180.0\mathrm{MeV}{\mathrm{fm}}^3$ and (b) $G_{pn}^{\mathrm{IS}}=0$ was used. The inset is a magnification of a high-energy region. Panel (a) was taken from Ref. [39].

Figure 8

GT transition strength from ${}^{136}\mathrm{Xe}$ to ${}^{136}\mathrm{Cs}$ in a low-energy region; (a) experimental data deduced from cross section of (${}^3\mathrm{He},t$) reaction [50], (b) calculated GT strength with $G_{pn}^{\mathrm{IS}}=-55.0\mathrm{MeV}{\mathrm{fm}}^3$, and (c) the same as panel (b) but for $G_{pn}^{\mathrm{IS}}=0$. Panel (b) was taken from Ref. [51].