Isospin-mixing and isospin-breaking corrections to superallowed $\beta$-decay

Figure 2: Isospin impurities in the ground states of $^{40}$Ca (upper panel) and $^{100}$Sn (lower panel), plotted as functions of the excitation energy of the doorway state for a set of commonly used Skyrme EDFs.[16] Results of the linear fits and the corresponding regression coefficients, $R$, are also shown.

Evaluation of $\alpha_C$ is a prerequisite to calculate isospin corrections to reaction and decay rates. As is well known,[17] isospin impurities are the largest in $N=Z$ nuclei, increase along the $N=Z$ line with increasing proton number, and are strongly quenched with increasing $\vert T_z\vert=\vert N-Z\vert/2$. Such characteristics were also early estimated based on the perturbation theory[18] or hydrodynamical model.[19] Quantitatively, after getting rid of the spurious mixing, which lowers the true $\alpha_C$ by as much as 30%,[7] the isospin impurity increases from a fraction of a percent in very light $N=Z$ nuclei to $\sim$0.9% in $^{40}$Ca, and $\sim$6.0% in $^{100}$Sn, as shown in Fig. 2. In the particular case of $^{80}$Zr, the calculated impurity of 4.4% agrees well with the empirical value deduced from the giant dipole resonance $\gamma$-decay studies.[20] This makes us believe that our model is indeed capable of capturing essential physics associated with the isospin mixing. Unfortunately, current experimental errors are too large to discriminate between different parametrizations of the Skyrme functional. The variations between EDFs in Fig. 2 result in $\sim$10% uncertainty in calculated values of $\alpha_C$.

The magnitude of theoretical $\alpha_C$ is quite well correlated with the excitation energy, $E_{T=1}$, of the $T=1$ doorway state, see Fig. 2. However, in order to make a precise determination of $E_{T=1}$, spectroscopic quality EDFs are needed, and this is not yet the case.[21] This explains why the values of $\alpha_C$ do not correlate well with basic EDF characteristics, including the isovector and isoscalar effective mass, symmetry energy, binding energy per particle, and incompressibility (see discussion in Ref.[9]).

Figure 3: Values of $\vert V_{ud}\vert$ deduced from the superallowed $\beta$-decay (full circles) for three different sets of the $\delta_C$ corrections calculated in: Ref.[2] (a); Ref.[22] with NL3 and DD-ME2 Lagrangians (b); and in the present work (c). Triangles mark values of $\vert V_{ud}\vert$ obtained from the pion-decay[23] and neutron-decay[24] studies, respectively. The open circle shows the value deduced from the $\beta$-transitions in $T=1/2$ mirror nuclei.[25]

Increasing demand on precise values of isospin impurities has been stimulated by the recent high-precision measurements of superallowed $\beta$-decay rates.[26,2] A reliable determination of the corresponding isospin-breaking correction, $\delta_C$, requires the isospin- and angular-momentum-projected DFT.[9]. This correction is obtained by calculating the $0^+ \rightarrow 0^+$ Fermi matrix element of the isospin raising/lowering operator $\hat T_{\pm}$ between the ground state (g.s.) of the even-even nucleus $\vert I=0, T\approx 1, T_z = \pm 1 \rangle$ and its isospin-analogue partner in the $N=Z$ odd-odd nucleus, $\vert I=0, T\approx 1, T_z = 0 \rangle$:

$$\vert\langle I=0, T\approx 1, T_z = \pm 1 \vert \hat T_{\pm}... ..., T\approx 1, T_z = 0 \rangle \vert^2 \equiv 2 ( 1-\delta_C ).$$ (4)

To determine the $\vert I=0, T\approx 1, T_z = 0 \rangle$ state in the odd-odd $N=Z$ nucleus, we first 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) HF orbits. Then, to correct for the fact that the antialigned configurations manifestly break the isospin symmetry,[8] that is, $\vert\bar \nu \otimes \pi \rangle \approx \frac{1}{\sqrt 2} (\vert T=0 \rangle + \vert T=1 \rangle )$, we apply the isospin and angular-momentum projections to create the basis $\vert I,M,K,T,T_z=0 \rangle$, in which the total Hamiltonian is rediagonalized (see Sec. 2). A similar scheme is used to compute the $\vert I=0, T\approx 1, T_z = \pm 1 \rangle$ states in the even-even nuclei.

Our studies indicate[9] that to obtain a fair estimate of $\delta_C$ for $A<40$ and $A>40$ nuclei, one needs to use large harmonic oscillator bases consisting of at least $N=10$ and 12 full shells, respectively. Even then, the results are subject to systematic errors due to the basis cut-off, which can be estimated to be $\sim$10%. Despite the fact that not all $N=12$ calculations in heavy ($A>40$) nuclei have yet been completed, and that owing to the shape-coexistence effects, there are still some ambiguities concerning the global minima, our preliminary results point to encouraging conclusions. Namely, the mean value of the structure-independent statistical-rate function $\bar{{\cal F}}t$,[26] obtained for 12 out of 13 transitions known empirically with high precision (excluding the $^{38}$K$\rightarrow$$^{38}$Ar case), equals $\bar{{\cal F}}t = 3069.4(10)$, which gives the value of the CKM matrix element equal to $\vert V_{ud}\vert= 0.97463(24)$. These values match well those obtained by Towner and Hardy in their recent compilation[2] (see Fig. 3). Because of a poor spectroscopic quality of the SV parameterization, the confidence level[27] of our results is poor. Nevertheless, it should be stressed that our method is quantum-mechanically consistent (see discussion in Refs.[28,29]) and contains no adjustable free parameters.