Coulomb rediagonalization and the isospin mixing

To properly account for the isospin mixing effects, following Refs. [28,29], the total Hamiltonian ${\hat H}_{NN}$ (5) (strong interaction plus the Coulomb interaction) is rediagonalized in the space spanned by the good-isospin wave functions (1), and the resulting eigenstates are denoted by

$$\vert n,T_z\rangle = \sum_{T\geq \vert T_z\vert}a^n_{T,T_z}\vert T,T_z\rangle$$ (59)

and numbered by index $n$. The amplitudes $a^n_{T,T_z}$ define the degree of isospin mixing. In particular, the isospin-mixing parameter for the lowest energy solution $E_{n=1,T_z}$ is defined as $\alpha_C = 1- \vert a^{n=1}_{\vert T_z\vert,T_z}\vert^2$.

Precise determination of the Coulomb mixing constitutes a notoriously difficult problem, but is strongly motivated by its impact on fundamental physics tested through the super-allowed $\beta$ decays [50,51,52]. Of importance here is to capture the proper balance between the short-range strong interaction and the long-range Coulomb polarization. This balance is naturally taken into account by the DFT approach, but is not accessible within a perturbative-analysis theory [53] or hydrodynamical model [54]. This is illustrated in Fig. 3 that shows the excitation energies of the doorway $T$=1 states in e-e $N$=$Z$ nuclei, i.e., the energies of the $n=2$ states in Eq. (59) relative to the HF g.s. energies $E_{HF}$. The estimates given by the perturbative/hydrodynamical approaches are clearly at variance with the self-consistent results.

Figure: Excitation energies of the doorway $T$=1 states in the e-e $N$=$Z$ nuclei relative to the SHF g.s. energies $E_{HF}$. Diamonds, dots, and circles show the self-consistent results obtained in the isospin-projected DFT with SIII [41], SLy4 [55], and SkP [56] Skyrme functionals, respectively. Horizontal lines mark the mean DFT values. Thick dotted lines show estimates based on the perturbation theory, $E_{T=1}\approx 82/A^{1/3}$MeV [53,54], and on the hydrodynamical model, $E_{T=1}\approx 169/A^{1/3}$MeV [54].

Figure 3 shows that not only the values but also the $A$-dependence of the doorway excitation energies differ substantially from the self-consistent results, pointing very clearly to a non-perturbative origin of the Coulomb mixing. One can also notice that the mean excitation energies of the doorway states (directly impacting the Coulomb mixing) strongly depend on the EDF parameterization. The currently used Skyrme functionals are not sufficiently constrained in the isospin sector to provide reliable estimates of $\alpha_C$. At this point is also not at all obvious what parts of EDF have to be refined in order to improve on this situation, see Ref. [29] for further details.