The choice of Skyrme interaction

As discussed in Ref. [24], the isospin projection technique outlined above does not yield singularities in energy kernels; hence, it can be safely executed with all commonly used energy density functionals (EDFs). However, as demonstrated in Ref. [15], the isospin projection alone leads to unphysically large isospin mixing in odd-odd $N=Z$ nuclei. It has thus been concluded that - in order to obtain reasonable results - isospin projection must be augmented by angular-momentum projection. This not only increases the numerical effort, but also brings back the singularities in the energy kernels [15] and thus prevents one from using the modern parametrizations of the Skyrme EDFs, which all contain density-dependent terms [35]. Therefore, the only option [15] is to use the Hamiltonian-driven EDFs. For the Skyrme-type functionals, this leaves us with one choice: the SV parametrization [36]. In order to better control the time-odd fields, the standard SV parametrization must be augmented by the tensor terms, which were neglected in the original work [36].

This density-independent parameterization of the Skyrme functional has the isoscalar effective mass as low as $\frac{m^*}{m}\approx 0.38$, which is required to reproduce the actual nuclear saturation properties. The unusual saturation mechanism of SV has a dramatic impact on the overall spectroscopic quality of this force, impairing such key properties like the symmetry energy [15], level density, and level ordering. These deficiencies also affect the calculated isospin mixing, which is a prerequisite for realistic estimates of $\delta_{\rm C}$. In particular, in the case of $^{80}$Zr discussed above, SV yields $\alpha_{\rm C} \approx 2.8$%, which is considerably smaller than the mean value of $\bar\alpha_{\rm C} \approx 4.4\pm 0.3$% obtained by averaging over nine popular Skyrme EDFs including the MSk1, SkO', SkP, SLy4, SLy5, SLy7, SkM$^*$, SkXc, and SIII functionals, see Ref. [17] for further details. Even though the ISB corrections $\delta_{\rm C}$ are primarily sensitive to differences between isospin mixing in isobaric analogue states, the lack of a reasonable Hamiltonian-based Skyrme EDF is probably the most critical deficiency of the current formalism.

The aim of this study is (i) to provide the most reliable set of the ISB corrections that can be obtained within the current angular-momentum and isospin-projected single-reference DFT, and (ii) explore the sensitivity of results to EDF parameters, choice of particle-hole configurations, and structure of time-odd fields that correlate valence neutron-proton pairs in odd-odd $N=Z$ nuclei. In particular, to quantify uncertainties related to the Skyrme coupling constants, we have developed a new density-independent variant of the Skyrme force dubbed hereafter SHZ2, see Table 1. The force was optimized purposefully to properties of light magic nuclei below $^{100}$Sn. It appears that the fit to light nuclei only weakly constrains the symmetry energy. The bulk symmetry energy of the SHZ2 is $a_{sym} \approx 42.2$MeV, that is, it overestimates the accepted value $a_{sym} \approx 32\pm 2$MeV by almost 30%. While this property essentially precludes using SHZ2 in detailed nuclear structure studies, it also creates an interesting opportunity for investigating the quenching of ISB effects due to the large isospin-symmetry-restoring components of the force.

Table 1: Skyrme parameters $t_i, x_i$ ( $i = 0,1,2,3$), and $W$ of SV [36] (second column) and SHZ2 (third column). The last column shows relative changes of parameters (in percent). Both parametrizations use the nucleon-mass parameter of $\hbar^2/2m=20.73$MeVfm$^2$. Parameters not listed are equal to zero.

param.	SV	SHZ2	change (%)
$t_0$	$-$1248.290	$-$1244.98830	$-$0.26
$t_1$	970.560	970.01156	$-$0.06
$t_2$	107.220	99.50197	$-$7.20
$x_0$	$-$0.170	0.01906	$-$111.21
$W$	150	150	0