ISB correction to $^{38}$K$\rightarrow$$^{38}$Ca transition

In the static DFT calculations, the ISB correction to the $^{38}$K$\rightarrow$$^{38}$Ar and $^{38}$Ca$\rightarrow$$^{38}$K superallowed transitions turned out to be unphysically large [3], and were disregarded. The reason could be traced back to unphysical values of the single-particle (s.p.) energies of the $2s_{1/2}$ and $1d_{3/2}$ orbits, which, for the SV functional, in the double magic nucleus $^{40}$Ca are almost degenerate and can therefore strongly mix, in particular through the time-odd fields in odd-odd $^{38}$K. To gain a better insight into the problem, in this work we perform the NCCI study of both nuclei, $^{38}$K and $^{38}$Ca. For our preliminary results presented in this work, we were able to converge three low-lying antialigned reference configurations in $^{38}$K and four configurations in $^{38}$Ca. Their basic properties are listed in Table II.

Table: Properties of reference Slater determinants in $^{38}$K and $^{38}$Ca nuclei, including their excitation energies, valence particle alignments in odd-odd nucleus $^{38}$K and their orientations, quadrupole moments, and triaxiality. The determinants are labeled by Nilsson quantum numbers pertaining to dominant components of the hole states.

$k$	$\vert^{38}$ K$;k\rangle$	$\Delta$E$_{\rm HF}$	$j_\nu / j_\pi$	$Q_2$	$\gamma$	$\vert^{38}$Ca$;k\rangle$	$\Delta$E$_{\rm HF}$	$Q_2$	$\gamma$
		(MeV)		(fm$^2$)	($^\circ$)		(MeV)	(fm$^2$)	($^\circ$)
1	$\vert 202\tfrac{3}{2}\rangle^{-2}$	0.000	-0.50/0.50(Y)	0.44	60	$\vert 200\tfrac{1}{2}\rangle^{-2}$	0.000	0.47	60
2	$\vert 220\tfrac{1}{2}\rangle^{-2}$	1.380	0.50/-0.50(Z)	0.18	0	$\vert 200\tfrac{1}{2}\rangle^{-2}$	0.762	0.03	0
3	$\vert 211\tfrac{1}{2}\rangle^{-2}$	1.559	-1.50/1.50(Z)	0.22	0	$\vert 211\tfrac{1}{2}\rangle^{-2}$	1.669	0.24	0
4						$\vert 220\tfrac{1}{2}\rangle^{-1}\otimes\vert 202\tfrac{3}{2}\rangle^{-1}$	2.903	0.09	60

Results of our NCCI calculations, giving energies of the $0^+$ states and the corresponding ISB corrections to $\beta$-decays, are visualized in Fig. 3. Again, our model accurately reproduces the experimental excitation energy of the second $0^+_2$ state in $^{38}$Ca. Indeed, the measured value, $\Delta E_{\rm EXP}=3057(18)$keV, is only 186keV higher than the calculated one, $\Delta E_{\rm TH}=2871$keV. The ISB corrections to the $^{38}$Ca$\rightarrow$$^{38}$K transitions are for $0_1^+\rightarrow 0_1^+$ and $0_2^+\rightarrow 0_2^+$ equal to 1.7% and 1.5%, respectively. As compared to the static theory, which for the $0_1^+$ states gives $\delta_{\rm C}$=8.9%, these values are strongly reduced, but they are almost twice larger than the result of TH [11], who quote $\delta_{\rm C}$=0.745(70)%.

Let us finally mention that the calculated energies of $0_2^+$ relative to $0_1^+$ states in $^{38}$K and $^{38}$Ar (preliminary value resulting from mixing of $0^+$ states projected from three HF configurations) are $\Delta E_{\rm TH}=2757$keV and $\Delta E_{\rm TH}=3161$keV, respectively. The latter value is in very good agreement with the experimental relative energy equal to $\Delta E_{\rm EXP}=3377.45(12)$keV.