Introduction

The isospin-symmetry violation in atomic nuclei is predominantly due to the Coulomb interaction that exerts long-range polarizations on neutron and proton states. To consistently take into account this polarization, one needs to employ huge configuration spaces. For that reason, an accurate description of isospin impurities in atomic nuclei, which is strongly motivated by the recent high-precision measurements of the $0^+ \rightarrow 0^+$ Fermi superallowed $\beta$-decay rates, is difficult to be obtained in shell-model approaches, and specific approximate methods are required.[1,2]

The long-range polarization effects can be included within the self-consistent mean-field (MF) or DFT approaches, which are practically the only microscopic frameworks available for heavy, open-shell nuclei with many valence particles. These approaches, however, apart from the physical contribution to the isospin mixing, mostly caused by the Coulomb field and, to a much lesser extent, by isospin-non-invariant components of the nucleon-nucleon force, also introduce the spurious isospin mixing due to the spontaneous isospin-symmetry breaking.[3,4,5]

Hereby, we present results on the isospin mixing and isospin symmetry-breaking corrections to the superallowed Fermi $\beta$-decay obtained by using the newly developed isospin- and angular-momentum-projected DFT approach without pairing.[6,7,8,9] The model employs symmetry-restoration techniques to remove the spurious isospin components and restore angular momentum symmetry, and takes advantage of the natural ability of MF to describe self-consistently the subtle balance between the Coulomb force making proton and neutron wave functions different and the isoscalar part of the strong interaction producing the opposite effect.

The paper is organized as follows. In Sec. 2, we describe the main theoretical building blocks of the isospin- and angular-momentum-projected DFT. Section 3 presents some preliminary applications of the formalism to the isospin symmetry-breaking corrections to the Fermi superallowed $\beta$-decay matrix elements, whereas Sec. 4 discusses applications of the isospin-projected DFT to nuclear symmetry energy. The summary is contained in Sec. 5.