Introduction

Atomic nucleus is a self-bound finite system composed of neutrons and protons that interact by means of short-range, predominantly isospin-symmetry-conserving strong force and long-range isospin-symmetry-breaking Coulomb force. In studies of phenomena related to the isospin-symmetry violation in nuclei, capturing a delicate balance between these two forces is of utmost importance. This is particularly true when evaluating the isospin-symmetry-breaking (ISB) corrections to superallowed $\beta$-decays between isobaric analogue states, [ $I=0^+,T=1]\longrightarrow [I=0^+,T=1]$.

Such $\beta$-decays currently offer the most precise data that give estimates of the vector coupling constant $G_V$ and leading element $V_{ud}$ of the Cabibbo-Kobayashi-Maskawa (CKM) flavor-mixing matrix [1,2]. The uncertainty of $V_{ud}$ extracted from the superallowed $\beta$-decays is almost an order of magnitude smaller than that from neutron or pion decays [3]. To test the weak-interaction flavor-mixing sector of the Standard Model of elementary particles, such precision is critical, because it allows us to verify the unitarity of the CKM matrix, violation of which may signal new physics beyond the Standard Model, see Ref. [4] and references cited therein.

The isospin impurity of the nuclear wave function - a measure of the ISB - is small. It varies from a fraction of a percent, in ground states of even-even $N=Z$ light nuclei, to about six percent in the heaviest known $N=Z$ system, $^{100}$Sn [5]. Nevertheless, its microscopic calculation poses a real challenge to theory. The reason is that the isospin impurity originates from the long-range Coulomb force that polarizes the entire nucleus and can be, therefore, calculated only within so-called no-core approaches. In medium and heavy nuclei, it narrows the possible microscopic models to those rooted within the nuclear density functional theory (DFT) [6,7].

The absence of external binding requires that the nuclear DFT be formulated in terms of intrinsic, and not laboratory densities. This, in turn, leads to the spontaneous breaking of fundamental symmetries of the nuclear Hamiltonian, including the rotational and isospin symmetries, which in finite systems must be restored. Fully quantal calculations of observables, such as matrix elements of electromagnetic transitions or $\beta$-decay rates, require symmetry restoration. In most of practical applications, this is performed with the aid of the generalized Wick's theorem [8]. Its use, however, leads to the energy density functionals (EDFs) being expressed in terms of the so-called transition densities, that is, to a multi-reference (MR) DFT. Unfortunately, the resulting MR EDFs are, in general, singular and require regularization, which still lacks satisfactory and practical solution, see, e.g., Refs. [9,10,11]. An alternative way of building a non-singular MR theory, the one that we use in the present work, relies on employing the EDFs derived from a true interaction, which then acquires a role of the EDF generator [12]. The results presented here were obtained using in this role the density-independent Skyrme interaction SV [13], augmented by the tensor terms (SV$_T$) [11].

Over the last few years we have developed the MR DFT approach based on the angular-momentum and/or isospin projections of single Slater determinants. The model, below referred to as static, was specifically designed to treat rigorously the conserved rotational symmetry and, at the same time, tackle the explicit Coulomb-force mixing of good-isospin states. These unique approach allowed us to determine the isospin impurities in $N\approx Z$ nuclei [5] and ISB corrections to superallowed $\beta$-decay matrix elements [14,15].

In this paper, following upon preliminary results announced at several conferences [16,17,18], we introduce a next-generation dynamic variant of the approach, which we call no-core configuration-interaction (NCCI) model. It constitutes a natural extension of the static MR DFT model, and allows for mixing states that are projected from different self-consistent Slater determinants representing low-lying (multi)particle-(multi)hole excitations. Technically, the model is analogous to the generator-coordinate-method (GCM) mixing of symmetry-projected states, see, e.g., Ref. [19]. However, the GCM pertains to mixing continuous sets of states, and thus builds collective states of the system, whereas NCCI involves mixing of discrete configurations. In quantum chemistry such a method is commonly known under the name of configuration interaction (CI), where the interaction means mixing of different electronic configurations. In nuclear physics, models of this type go by the name of the shell model, whereupon all configurations within a specific valence shell are considered. In recent years, in relatively light nuclei, a no-core variant of the shell model (NCSM) has been very successfully implemented [20]. Our approach combines the no-core aspect of the NCSM and the mixing aspect of the CI, and, by using sets of selected DFT configurations, it is not limited to light nuclei.

There are several cases when, to perform reliable calculations, the NCCI approach is indispensable. One of the most important ones relates to different possible shape-current orientations, which within the static variant of the model appear in odd-odd nuclei [15]. The configuration mixing is also needed to resolve the issue of unphysical ISB corrections to the analogous states of the $A=38$ isospin triplet [14,15].

The states that are mixed have good angular momenta and, at the same time, include properly evaluated Coulomb isospin mixing; hence, the extended model treats hadronic and Coulomb interactions on the same footing. The model is based on a truncation scheme dictated by the self-consistent deformed Hartree-Fock (HF) solutions, and can be used to calculate spectra, transitions, and $\beta$-decay rates in any nucleus, irrespective of its even or odd neutron and proton numbers.

We begin by giving in Sec. 2 a short overview of the theoretical framework of our NCCI model. In Sec. 3, a new set of the ISB corrections to the canonical set of superallowed $\beta$-decay is presented. As compared to our previous results [15], the new set includes mixing of reference states corresponding to different shape-current orientations in odd-odd $N=Z$ nuclei. In Sec. 4, applications involving mixing of several low-energy (multi)particle-(multi)hole excitations are discussed. Here, we determined low-spin energy spectra in selected nuclei relevant to high-precision tests of the weak-interaction flavor-mixing sector of the Standard Model. The calculations were performed for: $^{6}$Li and $^{8}$Li nuclei (Sec. 4.1), $A$=38 Ar, K, and Ca nuclei (Sec. 4.2), $^{42}$Sc and $^{42}$Ca nuclei (Sec. 4.3), and $^{62}$Ga and $^{62}$Zn nuclei (Sec. 4.4. Summary and perspectives are given in Sec. 5.