Confidence level test

In this section, we present results of the confidence-level (CL) test proposed in Ref. [5]. The CL test is based on the assumption that the CVC hypothesis is valid up to at least $\pm 0.03$%, which implies that a set of structure-dependent corrections should produce statistically consistent set of ${\cal F}t$-values. Assuming the validity of the calculated corrections $\delta_{\rm NS}$ [7], the empirical ISB corrections can be defined as:

$$\delta_{\rm C}^{{\rm (exp)}} = 1 + \delta_{\rm NS} - \frac{\overline{{\cal F}t}}{ft(1+\delta_{\rm R}^\prime)}.$$ (23)

By the least-square minimization of the appropriate $\chi^2$, and treating the value of $\overline{{\cal F}t}$ as a single adjustable parameter, one can attempt to bring the set of empirical values $\delta_{\rm C}^{{\rm (exp)}}$ as close as possible to the set of $\delta_{\rm C}$.

Figure 11: Top: differences between the calculated ISB corrections and empirical values resulting from the CL test of Ref. [5]. The shaded area of width $\pm 0.2$% is added in order to better visualize the differences. Bottom: contributions from individual transitions to the $\chi^2$ budget. Note the particularly large contributions from the $^{34}$Cl $\rightarrow ^{34}$S and $^{62}$Ga $\rightarrow ^{62}$Zn transitions that deteriorate the CL test. See text for details.

The empirical ISB corrections deduced in this way are tabulated in Table 2 and illustrated in Fig. 11. Table 2 also lists individual contributions to the $\chi^2$ budget. The obtained $\chi^2$ per degree of freedom ($n_d=11$) is $\chi^2 /n_d = 10.2$. This number is twice as large as that quoted in our previous work [16], because of the large uncertainty of $\delta_{\rm C}$ for the $^{34}$Cl $\rightarrow ^{34}$S transition. Other than that, both previous and present calculations have difficulty in reproducing the strong increase for $A=62$. Our $\chi^2 /n_d$ is also higher than the perturbative-model values reported in Ref. [5] ( $\chi^2 /n_d = 1.5$), shell model with Woods-Saxon (SM-WS) radial wave functions (0.4) [3], shell model with Hartree-Fock (SM-HF) radial wave functions (2.0) [45,4], Skyrme-Hartree-Fock with RPA (2.1) [12] , and relativistic Hartree-Fock plus RPA model (RHF-RPA) [13], which yields $\chi^2 /n_d = 1.7$.