An accurate evaluation of is a prerequisite for
determining
the isospinbreaking correction to the
Fermi
matrix element of the isospin raising/lowering operator
between nuclear states connected by the superallowed decay:


Restoration of angular momentum turns out to be the key ingredient in evaluation of the isospin impurity in oddodd nuclei. This is illustrated in Fig. 3, which shows calculated for the states in Sc. Four solutions shown in Fig. 3 correspond to the four possible antialigned MF configurations built on the Nilsson orbits originating from the spherical and subshells. These configurations can be labeled in terms of the quantum numbers, , 3/2, 5/2, and 7/2, as . In a simple shellmodel picture, each of those MF states contains all =0, 2, 4, and 6 components. From the results shown in Fig. 3, it is evident that the isospin projection alone (upper panel) leads to unphysically large impurities, whereas the impurities obtained after the isospin and angularmomentum (=0) projection (lower panel with the scale expanded by the factor of 500) are essentially independent of the initial MF configuration, as expected. The average value and standard deviation of 0.586(2)% shown in the figure were obtained for the configuration space of spherical harmonic oscillator (HO) shells, whereas for the analogous result is 0.620(2)% (see below).
Although indispensable, the angularmomentum projection creates numerous practical difficulties when applied in the context of DFT, that is, with energy functional rather than Hamiltonian. The major problem is the presence of singularities in energy kernels [28]. Although appropriate regularization schemes have already been proposed [29], they have neither been tested nor implemented. This fact narrows the applicability of the model only to those EDF parametrizations which strictly correspond to an interaction, wherefore the singularities do not appear. For Skyrmetype functionals, this leaves only one EDF parametrization, namely SV [30]. This specific EDF contains no density dependence and, after including all tensor terms in both timeeven and timeodd channels, it can be related to a twobody interaction. Despite the fact that for basic observables and characteristics such as binding energies, level densities, and symmetry energy, SV performs poorly, we have decided to use it in our systematic calculations of . Indeed, while SV would not be our first choice for nuclear structure predictions, it is still expected to capture essential polarization effects due to the selfconsistent balance between the longrange Coulomb and shortrange nuclear forces.
In order to test the performance of our model, we have selected the superallowed decay transition O N. This case is particularly simple, because (i) the participating nuclei are spherical and almost doubly magic, which implies suppressed pairing correlations, and (ii) the antialigned configuration in N involves a single configuration that is uniquely defined. The predicted values of are shown in Fig. 4 as a function of the assumed configuration space (that is, the number of spherical HO shells used). While the full convergence has not yet been achieved, this result, taken together with other tests performed for heavier nuclei, suggests that at least shells are needed for light nuclei (), whereas at least shells are required for heavier nuclei. The resulting systematic error due to basis cutoff is estimated at the level of 10%.
Even though calculations for all heavy () nuclei of interest are yet to be completed, and due to the shapecoexistence effects there are still some ambiguities concerning the choice of global minima, our very preliminary results are encouraging. Namely, the mean value of the structureindependent statisticalrate function , obtained for 12 out of 13 transitions known empirically with high precision (excluding KAr case), equals , which gives the amplitude of the CKM matrix. These values match very well those obtained by Towner and Hardy in their latest compilation [7]. That said, owing to the poor quality of the SV parameterization, the confidence level [10] of our results is low. On a positive note, our method is quantum mechanically consistent (see discussion in Ref. [8]) and contains no free parameters.