Theoretical uncertainties and error analysis

Based on the discussion presented in Secs. 3.1 and 3.2, the recommended calculated values of $\delta_{\rm C}$ for the superallowed $0^+ \rightarrow 0^+$ $\beta$-decay are determined by averaging over three relative orientations of shapes and currents. Only in the case of $A=42$, we adopt for $\delta_{\rm C}$ an arithmetic mean over the four configurations associated with different $K$-orbitals.

To minimize uncertainties in $\alpha_{\rm C}$ and $\delta_{\rm C}$ associated with the truncation of HO basis in HFODD, we used different HO spaces in different mass regions, cf. Sec. 2.3. With this choice, the resulting systematic errors due the basis cut-off should not exceed $\sim 10$%. To illustrate the dependence of $\delta_{\rm C}$ on the number of HO shells, Fig. 5 shows the case of the superallowed $^{46}$V $\rightarrow$$^{46}$Ti transition obtained by projecting from the $\vert\varphi_{\rm Z}\rangle$ solution in $^{46}$V. In this case, the parent and daughter nuclei are axial, which allows us to reduce the angular-momentum projection to one-dimension and extend the basis size up to $N=20$ HO shells.

With increasing $N$, $\delta_{\rm C}$ increases, and asymptotically it reaches the value of $0.8096(12)$%. This limiting value is about 6.7% larger than the value of $0.7587$% obtained for $N=12$ shells, that is, for a basis used to compute the $42 \leq A\leq 54$ cases. For $62\leq A\leq 74$ nuclei, which were all found to be triaxial, we have used $N=14$ shells. The further increase of basis size is practically impossible. Nonetheless, as seen in Fig. 5, a rate of increase of $\delta_{\rm C}$ slows down exponentially with $N$, which supports our 10% error estimate due to the basis truncation.

Figure 5: (Color online) The convergence of the ISB correction $\delta_{\rm C}$ to the $^{46}$V $\rightarrow$$^{46}$Ti superallowed $\beta$-decay versus the number of oscillator shells considered.

The total error of the calculated value of $\delta_{\rm C}$ includes the standard deviation from the averaging, $\sigma_n$, and the assumed 10% uncertainty due to the basis size: $\Delta (\delta_{\rm C}) = \sqrt{\sigma_n^2 + (0.1\delta_{\rm C})^2 }$. The same prescription for $\Delta (\delta_{\rm C})$ was also used in the test calculations with SHZ2, even though a slightly smaller HO basis was employed in that case.

For $A=38$ nuclei, our model predicts the unusually large correction $\delta_{\rm C}\approx 10$%. The origin of a very different isospin mixing obtained for odd-odd and even-even members of this isobaric triplet is not fully understood. Most likely, it is a consequence of the poor spectroscopic properties of SV. Indeed, as a result of an incorrect balance between the spin-orbit and tensor terms in SV, the $2s_{1/2}$ subshell is shifted up in energy close to the Fermi surface. This state is more sensitive to time-odd polarizations than other s.p. states around $^{40}$Ca core, see Table I in Ref. [38]. The calculated equilibrium deformations $(\beta_2, \gamma)$ of the $T_z\pm 1$ and $T_z=0$ $A=38$ isobaric triplet are very similar, around (0.090, 60$^\circ$). In the following, the $^{38}$K $\rightarrow$$^{38}$Ar transition is excluded from the calculation of the $V_{\rm ud}$ matrix element.