Summary

In summary, the isospin- and angular-momentum-projected DFT calculations have been performed to estimate the isospin-breaking corrections to $0^+ \rightarrow 0^+$ Fermi superallowed $\beta$-decays. Preliminary results for the average value of the nucleus-independent $\bar{\cal F}t = 3069.4(10)$ and the amplitude $\vert V_{ud}\vert= 0.97463(24)$ were found to be consistent with the recent estimates by Towner and Hardy,[2] notwithstanding a low spectroscopic quality of the Skyrme EDF SV used.

Applicability of the isospin-projected DFT to analyze the nuclear symmetry energy has also been discussed. It has been demonstrated that the isospin projection offers a rather unique opportunity to study the interaction part of the symmetry energy in the odd-odd $N=Z$ nuclei and that this quantity is influenced by time-odd fields of the energy density functional.

This work was supported in part by the Polish Ministry of Science under Contract Nos. N N202 328234 and N N202 239037, 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-FC02-09ER41583 (UNEDF SciDAC Collaboration). We acknowledge the CSC - IT Center for Science Ltd, Finland for the allocation of computational resources.