$0^+$ states in $^{38}$Ca, $^{38}$K, and $^{38}$Ar

Recently, Park et al. [50] performed high-precision measurement of the superallowed $0^+\longrightarrow 0^+$ Fermi decay of $^{38}$Ca $\rightarrow$$^{38}$K, see also [51]. The reported $ft$ value of 3062.3(68)s was measured with a relative precision of $\pm$0.2%, which is sufficient for testing and determining the parameters of electroweak sector of the Standard Model. This piece of data is the first, after almost a decade, addition to a set of canonical $0^+\longrightarrow 0^+$ Fermi transitions, which are used to determine $\vert V_{ud}\vert$. Moreover, being a mirror partner to superallowed $0^+\longrightarrow 0^+$ Fermi transition $^{38}$K $\rightarrow$$^{38}$Ar, it allows for sensitive tests of the ISB corrections and, in turn, for assessing quality of nuclear models used to compute the ISBs [50].

Unfortunately, using the DFT with the SV Skyrme functional, which gives a strong mixing between the $2s_{1/2}$ and $1d_{3/2}$ orbits, it is difficult to determine the ISB corrections to the $^{38}$K $\rightarrow$$^{38}$Ar and $^{38}$Ca $\rightarrow$$^{38}$K superallowed transitions. In particular, in our previous static DFT calculations, the ISB corrections turned out to be of the order of 9%, and thus were disregarded [14,15].

In Ref. [18], we presented preliminary results of the NCCI study of $^{38}$Ca and $^{38}$K. Here we extend them to calculations that include three low-lying antialigned reference configurations in $^{38}$K and four configurations in both $^{38}$Ca and $^{38}$Ar. Basic properties of these reference states are listed in Table 4.

Table 4: Similar as in Table 3, but for $^{38}$K, $^{38}$Ca, and $^{38}$Ar. Here, the reference Slater determinants are labeled by the Nilsson quantum numbers pertaining to dominant components of the hole states below $^{40}$Ca. The first excited state in $^{38}$Ar, marked by asterisk, was converged with a weak quadrupole constraint.

$i$	$\vert^{38}$K$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$j_\nu$	$j_\pi$	$k$
1	$\vert 202\tfrac{3}{2}\rangle^{-2}$	0.000	0.083	60$^\circ$	$-$0.50	0.50	Y
2	$\vert 220\tfrac{1}{2}\rangle^{-2}$	1.380	0.035	0$^\circ$	0.50	$-$0.50	Z
3	$\vert 211\tfrac{1}{2}\rangle^{-2}$	1.559	0.042	0$^\circ$	$-$1.50	1.50	Z
$i$	$\vert^{38}$Ca$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$j_\nu$	$j_\pi$	$k$
1	$\vert 200\tfrac{1}{2}\rangle^{-2}$	0.000	0.088	60$^\circ$	0	0	-
2	$\vert 200\tfrac{1}{2}\rangle^{-2}$	0.762	0.006	0$^\circ$	0	0	-
3	$\vert 211\tfrac{1}{2}\rangle^{-2}$	1.669	0.045	0$^\circ$	0	0	-
4	$\vert 220\tfrac{1}{2}\rangle^{-1}\otimes\vert 202\tfrac{3}{2}\rangle^{-1}$	2.903	0.015	60$^\circ$	0	0	-
$i$	$\vert^{38}$Ar$;i\rangle$	$\Delta$E$_{\rm HF}$	$\beta _2$	$\gamma$	$j_\nu$	$j_\pi$	$k$
1	$\vert 200\tfrac{1}{2}\rangle^{-2}$	0.000	0.088	60$^\circ$	0	0	-
2	$\vert 200\tfrac{1}{2}\rangle^{-2}$	0.651$^{(*)}$	0.002	46$^\circ$	0	0	-
3	$\vert 211\tfrac{1}{2}\rangle^{-2}$	1.600	0.045	0$^\circ$	0	0	-
4	$\vert 220\tfrac{1}{2}\rangle^{-1}\otimes\vert 202\tfrac{3}{2}\rangle^{-1}$	2.754	0.017	60$^\circ$	0	0	-

Results of our NCCI calculations, including the binding energies of the lowest $0_1^+$ states, excitation energies of the first excited $0_2^+$ states, and the ISB corrections to superallowed $\beta$-decays, are visualized in Fig. 4. The total binding energies of the $0_1^+$ states in these three nuclei are underestimated by circa 1%. Concerning the first excited $0_2^+$ states, our model works very well in $^{38}$Ca. In this nucleus, the measured excitation energy, $\Delta E_{\rm EXP}=3057(18)$keV, is only 186keV larger than the calculated one, $\Delta E_{\rm TH}=2871$keV. Note, however, that the calculated excitation energies of the $0_2^+$ states are predicted to decrease with increasing $T_z$, at variance with the data. In turn, the difference between experimental and theoretical excitation energies of the $0_2^+$ in $^{38}$Ar grows to approximately 0.7MeV.

The ISB corrections $\delta _{\rm C}$ to the $^{38}$Ca $\rightarrow$$^{38}$K transitions between the $0_1^+\rightarrow 0_1^+$ and $0_2^+\rightarrow 0_2^+$ states are equal to 1.7% and 1.5%, respectively. As compared to our previous static model, which for the $0_1^+$ states was giving an unacceptably large correction of 8.9%, the NCCI result is strongly reduced. Nevertheless, it is still almost twice larger than that of Towner and Hardy [27], who quote the value of 0.77(7)%.

Similar results were obtained for the $^{38}$K $\rightarrow$$^{38}$Ar transitions, where the calculated corrections are 1.3% ( $0_1^+\rightarrow 0_1^+$) and 1.4% ( $0_2^+\rightarrow 0_2^+$). Again, as compared to the static variant of our model, the value for the $0_1^+\rightarrow 0_1^+$ transition is strongly reduced, but it is considerably larger than the Towner and Hardy result of 0.66(6)%. Nevertheless, we see that the NCCI model removes, at lest partially, pathologies encountered in the static variant.

Figure 4: (a) Excitation energies of the $0_2^+$ states with respect to the $0_1^+$ states in the $A=38$ isobaric triplet nuclei: Ca, K, and Ar. Theoretical predictions and data [52] are shown with solid and dashed lines, respectively. Calculated values of $\delta _{\rm C}$ are also shown. Decays $0_2^+ \rightarrow 0_1^+$ indicated in the figure are predicted to be strongly hindered. (b) Comparison between the total binding energies of the $0_1^+$ states ($N=10$ harmonic oscillator shells were used).