Nearly degenerate $\bm {K}$-orbitals

Owing to an increased density of s.p. Nilsson levels in the vicinity of the Fermi surface for nearly spherical nuclei, there appears another type of ambiguity in choosing the Slater determinants representing the anti-aligned configurations. Within the set of nuclei studied in this work, this ambiguity manifests itself particularly strongly in $^{42}$Sc, where we deal with four possible anti-aligned MF configurations built on the Nilsson orbits originating from the spherical $\nu f_{7/2}$ and $\pi f_{7/2}$ sub-shells. These configurations can be labeled in terms of the quantum number $K$ as $\vert\nu\bar{K} \otimes \pi K \rangle$ with $K=1/2$, 3/2, 5/2, and 7/2.

In the extreme shell-model picture, each of these states contains all the $T=1$ and $I$=0, 2, 4, and 6 components. Within the projected DFT picture, owing to configuration-dependent polarizations in time-odd and time-even channels, the situation is more complicated because the Slater determinants $\vert\nu\bar{K} \otimes \pi K \rangle$ corresponding to different $K$-values are no longer degenerate. Consequently, for each angular momentum $I$, one obtains four different linearly-dependent solutions. Calculations show that in all $I$=0 and $T\approx 1$ states of interest, the isospin mixing $\alpha_{\rm C}$ is essentially independent of the choice of the initial Slater determinant. In contrast, the calculated ISB corrections $\delta_{\rm C}$ and energies depend on $K$, see Fig. 4.

Figure 4: Top: the ISB corrections to the $^{42}$Sc $\rightarrow$$^{42}$Ca superallowed $\beta$-transition, calculated using SV (circles) and SHZ2 (dots) forces by projecting the $\vert\nu\bar{K} \otimes \pi K \rangle$ configurations in $^{42}$Sc for $K=1/2$, 3/2, 5/2, and 7/2. From top to bottom, horizontal lines mark (i) the average ISB correction using SV (thick solid line); (ii) the value of Ref. [3] (dotted line); and (iii) $\delta_{\rm C}$ of Ref. [13]. Shaded regions mark the related uncertainties. Bottom: projected energies of states $\vert K; I=0^+, T\approx 1\rangle$ in $^{42}$Sc obtained from the configurations $\vert\nu\bar{K} \otimes \pi K \rangle$, relative to the projected energy of the $K=1/2$ state.