Isospin-breaking corrections to the Fermi matrix elements of the superallowed $\beta$-decay

An accurate evaluation of $\alpha_C$ is a prerequisite for determining the isospin-breaking correction $\delta_C$ to the $0^+\rightarrow 0^+$ Fermi matrix element of the isospin raising/lowering operator $\hat T_{\pm}$ between nuclear states connected by the superallowed $\beta$-decay:

$$\vert\langle I^\pi=0^+, T\approx 1, T_z = \pm 1 \vert \hat T... ..., T\approx 1, T_z = 0 \rangle \vert^2 \equiv 2 ( 1-\delta_C ).$$ (8)

Here, the state $\vert I^\pi=0^+, T\approx 1, T_z = \pm 1 \rangle$ corresponds to the g.s. of the even-even nucleus whereas $\vert I^\pi=0^+, T\approx 1, T_z = 0 \rangle$ denotes its isospin-analogue in the neighboring $N=Z$ odd-odd nucleus. Unlike the former one, the odd-odd configuration cannot be expressed in a form of a MF product wave function [13]. Therefore, to compute the states in odd-odd $N=Z$ nuclei, we use the following strategy (see Fig. 1 of Ref. [13] for a schematic illustration):

• Firstly, we compute the so-called antialigned g.s. configuration, $\vert\bar \nu \otimes \pi \rangle$ or $\vert \nu \otimes \bar \pi \rangle$, by placing the odd neutron and the odd proton in the lowest available time-reversed (or signature-reversed) single-particle Nilsson orbits.
• Secondly, to correct for the fact that the antialigned configurations manifestly break the isospin symmetry, that is, $\vert\bar \nu \otimes \pi \rangle \approx \frac{1}{\sqrt 2} (\vert T=0 \rangle + \vert T=1 \rangle )$, we apply the simultaneous isospin and angular-momentum projection to create the good isospin and good angular momentum basis $\vert I,M,K,T,T_z=0 \rangle$ of Eq. (6).
• Finally, to obtain the state $\vert I=0, T\approx 1, T_z = 0 \rangle$, we rediagonalize the total Hamiltonian, including the Coulomb term, in the new basis [cf. Eq. (7)].

The projected $\vert I^\pi=0^+, T\approx 1, T_z = \pm 1 \rangle$ states in even-even nuclei are computed in the same way.

Figure 3: Isospin impurities in $^{42}$Sc, calculated for four antialigned configurations that are obtained by putting the valence neutron and proton in opposite-$K$ Nilsson orbitals originating from the $f_{7/2}$ shell, that is, $\vert\nu\bar{K} \otimes \pi K \rangle$ with $K=1/2,3/2,5/2$, and 7/2. Open and full dots show the results obtained by employing only the isospin projection and simultaneous isospin and angular-momentum ($I=0$) projection, respectively.

Figure 4: Isospin-breaking correction to the Fermi matrix element for the superallowed transition $^{14}$O$\rightarrow$$^{14}$N. Full dots represent our results plotted as a function of the basis size (number of HO shells taken in HF calculations). A conservative $10$% error was assigned to the last two points. The values quoted in Refs. [7] (including errors) and [27] are shown for comparison.

Restoration of angular momentum turns out to be the key ingredient in evaluation of the isospin impurity in odd-odd nuclei. This is illustrated in Fig. 3, which shows $\alpha_C$ calculated for the $T\approx 1$ states in $^{42}$Sc. Four solutions shown in Fig. 3 correspond to the four possible antialigned MF configurations built on the Nilsson orbits originating from the spherical $\nu f_{7/2}$ and $\pi f_{7/2}$ subshells. These configurations can be labeled in terms of the $K$ quantum numbers, $K=1/2$, 3/2, 5/2, and 7/2, as $\vert\nu\bar{K} \otimes \pi K \rangle$. In a simple shell-model picture, each of those MF states contains all $I$=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 angular-momentum ($I$=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 $N=10$ spherical harmonic oscillator (HO) shells, whereas for $N=12$ the analogous result is 0.620(2)% (see below).

Although indispensable, the angular-momentum 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 Skyrme-type 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 time-even and time-odd channels, it can be related to a two-body 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 $\delta_C$. 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 self-consistent balance between the long-range Coulomb and short-range nuclear forces.

In order to test the performance of our model, we have selected the superallowed $\beta$-decay transition $^{14}$O $\longrightarrow$$^{14}$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 $^{14}$N involves a single $\vert\nu\bar{p}_{1/2} \otimes \pi p_{1/2} \rangle$ configuration that is uniquely defined. The predicted values of $\delta_C$ are shown in Fig. 4 as a function of the assumed configuration space (that is, the number of spherical HO shells $N$ 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 $N=10$ shells are needed for light nuclei ($A< 40$), whereas at least $N=12$ shells are required for heavier nuclei. The resulting systematic error due to basis cut-off is estimated at the level of $\sim$10%.

Even though calculations for all heavy ($A> 40$) nuclei of interest are yet to be completed, and due to the shape-coexistence effects there are still some ambiguities concerning the choice of global minima, our very preliminary results are encouraging. Namely, the mean value of the structure-independent statistical-rate function $\bar{{\cal F}}t$, obtained for 12 out of 13 transitions known empirically with high precision (excluding $^{38}$K$\rightarrow$$^{38}$Ar case), equals $\bar{{\cal F}}t = 3069.4(10)$, which gives the $V_{\mbox{\scriptsize {ud}}} = 0.97463(24)$ 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.