Summary and perspectives

In this work, we introduced the NCCI model involving the isospin and angular-momentum projections and subsequent mixing of states having good angular momentum and properly treated Coulomb isospin mixing. The model is capable of treating rigorously both the fundamental (spherical, particle-number) as well as approximate (isospin) nuclear symmetries. Its potentially unrestricted range of applicability and a natural ability to treat the core-polarization effects resulting from a subtle interplay between the long-range Coulomb force and short-range hadronic nucleon-nucleon forces, which are treated on the same footing, makes it an interesting alternative to the nuclear shell-model.

The NCCI model employs states projected from low-lying (multi)particle-(multi)hole deformed Slater determinants (configurations) calculated self-consistently using Hartree-Fock method. In the present realization, the same SV Skyrme functional was used both to compute the configurations and to mix the states. This restriction, however, can be easily relaxed opening a room for various generalizations of the model. In particular, one can attempt to correct an interaction used at the mixing stage in order to improve a description of $T=0,I=1^+$ states in $^6$Li and $^{42}$Sc $N=Z$ nuclei.

We demonstrated that our NCCI formalism is capable of capturing many features of the low-lying energy spectra in such diverse systems as $^{8}$Li, $A$=38 isospin triplet nuclei, or $^{62}$Zn and $^{62}$Ga nuclei. A reasonable agreement with experiment was obtained when using a relatively small number of configurations, which supports our claims that the model can indeed be applicable to medium heavy nuclei with an affordable numerical cost. Our recent systematic study of Gamow-Teller matrix elements in $T_z=1/2$ $sd$- and lower $pf$-shell mirror nuclei performed in Ref. [57], see also [63], confirms that the model can incorporate in a controlled way many important correlations into the nuclear wave function.

Finally, we also calculated the new set of the ISB corrections to superallowed $T=1,I=0^+\rightarrow T=1,I=0^+$ beta transitions. The refined corrections are collected in Table 1 for a canonical set of precisely measured transitions and in Table 2 for transitions that were either unmeasured or measured with the accuracy insufficient for the Standard Model tests. These results are based on mixing the $I=0^+$ states projected from the so-called $X$,$Y$, and $Z$ configurations corresponding to different shape-current orientations in odd-odd nuclei. Unfortunately, an attempt to perform more advanced calculation for the transition $^{62}$Ga $\rightarrow ^{62}$Zn, which would take into account more configurations, failed because of difficulties in matching the model spaces in even-even and odd-odd nuclei.

This work was supported in part by the Polish National Science Centre (NCN) under Contract Nos. 2012/07/B/ST2/03907 and 2014/15/N/ST2/03454, by the THEXO JRA within the EU-FP7-IA project ENSAR (No. 262010), by the ERANET-NuPNET grant SARFEN of the Polish National Centre for Research and Development (NCBiR), and by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme. We acknowledge the CSC-IT Center for Science Ltd., Finland, for the allocation of computational resources.