Introduction

Superallowed Fermi $\beta$-decays between the isobaric analogue states, [ $I=0^+,T=1]\rightarrow [I=0^+,T=1]$, provide the most precise values of the vector coupling constant $G_V$ and leading element $V_{ud}$ of the Cabibbo-Kobayashi-Maskawa (CKM) flavour-mixing matrix, which are critical for stringent tests of weak-interaction flavor-mixing sector of the Standard Model of particle physics. In particular, these data are needed for testing unitarity of the CKM matrix, violation of which may signalize new physics beyond the Standard Model, see [1] and refs. quoted therein.

In testing the Standard Model, precision is of utmost importance. Only the $\beta$-decays, for which the reduced life-times, $ft$, are measured with accuracy better than half a percent, can be used for that purpose. At present, thirteen such cases are known in nuclei, ranging in mass from $A$=10 to $A$=74. The extraction of $G_V$ and $V_{ud}$ is not solely dependent on experimental data but also requires theoretical input in the form of radiative and many-body corrections to the experimental $ft$ values. The corrections are small, of the order of a percent, but are critical for the applicability of the entire method, because it relies on the so-called conserved vector current hypothesis (CVC). The CVC hypothesis assumes independence of the vector current on nuclear medium, and must be verified a priori by investigating mass independence of the corrected reduced life-times defined as:

$${\cal F}t \equiv ft(1+\delta_{\rm R}^\prime)(1+\delta_{\r... ...m V}^2 (1 + \Delta^{\rm V}_{\rm R})} \approx {\rm const.}\, ,$$ (1)

where $K/(\hbar c)^6 = 2\pi^3 \hbar \ln 2 /(m_{\rm e} c^2)^5 = 8120.2787(11)\times 10^{-10}$GeV$^{-4}$s is a universal constant. Symbols $\delta_{\rm R}^\prime, \delta_{\rm NS}, \Delta^{\rm V}_{\rm R}$ are the radiative corrections while $\delta_{\rm C}$ stands for the isospin symmetry-breaking (ISB) correction to the Fermi matrix element.

Since the isospin symmetry is weakly broken, mostly by the Coulomb interaction that polarizes the entire nucleus, microscopic calculation of the ISB corrections is a challenging task. Capturing a delicate equilibrium between the hadronic and Coulomb effects is fully possible only within no core approaches. This, in heavier nuclei, reduces the possible choices to formalisms rooted in the density functional theory (DFT). However, as it was recognized already in the 70's [2], to determine the magnitude of isospin impurities, the self-consistent mean-field (MF) approaches cannot be directly applied, because of a spurious mixing caused by the spontaneous symmetry-breaking effects. This observation hindered theory from progress in the field for decades.

To overcome these problems, over the last few years we have developed a no-core multi-reference DFT, which involves the isospin- and angular-momentum projections of Slater determinants representing the $0^+$ triplet states in mother and daughter nuclei [3,4]. The formalism, dubbed static, was specifically designed to treat rigorously the conserved rotational symmetry and, at the same time, tackle the explicit breaking of the isospin symmetry due to the Coulomb field. Recently, by allowing for mixing of states that are projected from self-consistent Slater determinants representing low-lying (multi)particle-(multi)hole excitations, we have extended the model to the so-called dynamic variant [5]. The model belongs to the class of the no core configuration-interaction approaches, with the two-body short-range (hadronic) and long-range (Coulomb) interactions treated on the same footing. It is based on a truncation scheme dictated by the self-consistent deformed Hartree-Fock (HF) solutions. The model can be used to calculate spectra, transitions, and $\beta$-decay matrix elements in any nuclei, irrespective of their mass and neutron- and proton-number parities.

The aim of this work is to present this novel theoretical framework along with preliminary results for the low-spin spectra and $\beta$-decay matrix elements in selected $N\approx Z$ nuclei. The first applications of the model to the low-lying spectra in $^{32}$Cl and $^{32}$S have been published in [5]. Hereafter, we focus on nuclei relevant to high-precision tests of the weak-interaction flavor-mixing sector of the Standard Model. In this perspective, we discuss the spectrum of $0^+$ states in $^{62}$Zn, which was reassigned in a recent experiment [6], and is now posing a challenge to theory. We also briefly overview preliminary attempts and difficulties arising in determining the ISB correction to the superallowed $^{62}$Ga $\rightarrow ^{62}$Zn $0^+\rightarrow 0^+$ $\beta$-decay, which is strongly model dependent. We also present preliminary results for the ISB correction to the Fermi matrix element corresponding to the $^{38}$Ca $\rightarrow ^{38}$K transition. In our static calculations, the case of $A$=38 was excluded from the canonical pool of superallowed data. This was because of the anomalously large ISB correction, caused by uncontrolled mixing of the $2s_{1/2}$ and $1d_{3/2}$ orbits, which for the SV$_{\rm T}$ Skyrme true interaction are almost degenerate. The SV$_{\rm T}$ interaction is the SV functional [7] augmented with the tensor terms, see discussion in [8].

The paper is organized as follows. In Sec. 2, the basics of our dynamical model are briefly sketched. In Sec. 3, preliminary numerical results concerning spectrum of $0^+$ states in $^{62}$Zn and the ISB corrections for $^{38}$Ca$\rightarrow$$^{38}$K Fermi transitions are presented. The paper is summarized in Sec. 4.