Summary and perspectives

Within the recently-developed unpaired projected-DFT approach, we carried out systematic calculations of isospin mixing effects and ISB corrections to the superallowed ${0^+ \rightarrow 0^+}$ Fermi decays in $10 \le A \le 74$ nuclei and $\beta$-transitions between the isobaric analogue states in mirror ${T=1/2}$ nuclei with $11 \le A \le 49$. Our predictions are compared with empirical values and with predictions of other theoretical approaches. Using isospin-breaking corrections computed in our model, we show that the unitarity of the CKM matrix is satisfied with a precision better than 0.1%. We also provide ISB corrections for heavier nuclei with $78 \le A \le 98$ nuclei that can guide future experimental and theoretical studies.

We carefully analyze various model assumptions impacting theoretical uncertainties of our calculations: basis truncation, definition of the intrinsic state, and configuration selection. To assess the robustness of our results with respect to the choice of interaction, we compared SV results with predictions of the new force SHZ2 that has been specifically developed for this purpose. The comparison of SV and SHZ2 results suggest that ISB corrections are sensitive to the interplay between the bulk symmetry energy and time-odd mean-fields.

While the overall agreement with the empirical values offered by the projected-DFT approach is very encouraging, and the results are fairly robust, there is a lot of room for systematic improvements. The main disadvantages of our model in its present formulation include: (i) lack of pairing correlations; (ii) lack of ph interaction (or functional) of good spectroscopic quality; (iii) the use of a single HF reference state that cannot accommodate configuration mixing effects; (iv) ambiguities in establishing the HF reference state in odd and odd-odd nuclei caused by different possible orientations of time-odd currents with respect to total density distribution. The work on various enhancements of our model, including the inclusion of $T=0$ and $T=1$ pairing within the projected Hartree-Fock-Bogoliubov theory, better treatment of configuration mixing using the multi-reference DFT, and development of the spectroscopic-quality EDF used in projected calculations, is in progress.

This work was supported in part by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme, and by the Office of Nuclear Physics, U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee) and DE-SC0008499 (NUCLEI SciDAC-3 Collaboration). We acknowledge the CSC - IT Center for Science Ltd, Finland, for the allocation of computational resources.