especially for Sn, are calculated with D$^3$C, but these are still orders of magnitude from the experimental values. It appears that all three interactions reproduce the isotopic trend in the Te chain.

![(Color online) Same as in Fig. \[Fig2\], but for the Cd (left panel), Sn (middle panel), and Te (right panel) isotopic chains. For the D$^3$C effective interaction, in all three isotopic chains the strength of the $T=0$ pairing interaction is $V_0 = 235$ MeV (filled squares).[]{data-label="Fig3"}](Fig3.eps)

We have considered the effect of the $T=0$ pairing interaction on the calculated $\beta$-decay half-lives only for the D$^3$C effective interaction which, with the effective nucleon mass $m^*/m = 0.79$ comparable to those of non-relativistic effective interactions, gives the shortest half-lives. Even without the inclusion of the proton-neutron $pp$ interaction, for the Fe nuclei the calculated half-lives are already close to the experimental values, except for $^{64}$Fe (see Fig. \[Fig2\]). By adjusting the value of the strength parameter of the $T=0$ pairing to $V_0 = 125$ MeV, the PN-QRPA calculation reproduces the $\beta$-decay half-lives of $^{66}$Fe, $^{68}$Fe and $^{70}$Fe (filled squares). In the case of Ni isotopes the $T=0$ interaction in the $pp$-channel is not effective because of the $Z=28$ and $N=40$ closures [@Eng.99; @NMV.05]. The $\pi 1f_{7/2}$ orbit is completely occupied, and the transition $\nu 1f_{5/2} \to \pi 1f_{7/2}$ is thus blocked. The $T=0$ pairing could only have an effect on the $\nu 1g_{9/2} \to \pi 1g_{9/2}$ transition, but the $\pi 1g_{9/2}$ orbital is located high above the Fermi surface. Thus the $T=0$ $pp$ interaction cannot shift the low-energy GT strength and enhance the $\beta$-decay rates. Even using the D$^3$C interaction, the calculated half-lives are an order of magnitude longer than the experimental values.

The principal advantage of the self-consistent approach to the modeling of $\beta$-decay rates is the use of universal (A independent) interactions in the $ph$