Shape-current orientation

Figure 2: (Color online) Schematic illustration of relative orientations of shapes and currents in the three anti-aligned states $\vert\varphi_{\rm X}\rangle$ ($X$), $\vert\varphi_{\rm Y}\rangle$ ($Y$), and $\vert\varphi_{\rm Z}\rangle$ ($Z$) discussed in the text. The long ($Oz$), intermediate ($Ox$), and short ($Oy$) principal axes of the nuclear mass distribution are indicated by thick arrows. The odd-neutron ($j_\nu$) and odd-proton ($j_\pi$) angular momentum oriented along the $Ox$, $Oy$, or $Oz$ axes is shown by thin arrows. Note that in each case the total angular-momentum alignment, $j_\nu+j_\pi$, is zero.

At variance with the even-even parent nuclei, the anti-aligned configurations in odd-odd daughter nuclei are not uniquely defined. One of the reasons, which was not fully appreciated in our previous work [16], is related to the relative orientation of the nuclear shapes and currents associated with the valence neutron-proton pairs. In all signature-symmetry-restricted calculations for triaxial nuclei, such as ours, there are three anti-aligned Slater determinants with the s.p. angular momenta (alignments) of the valence protons and neutrons pointing, respectively, along the $Ox$, $Oy$, or $Oz$ axes of the intrinsic shape defined by means of the long ($Oz$), intermediate ($Ox$), and short ($Oy$) principal axes of the nuclear mass distribution. These solutions, hereafter referred to as $\vert\varphi_{\rm X}\rangle$, $\vert\varphi_{\rm Y}\rangle$, and $\vert\varphi_{\rm Z}\rangle$, are schematically illustrated in Fig. 2. Their properties can be summarized as follows:

• The three solutions are not linearly independent. Their Hartree-Fock (HF) binding energies may typically differ by a few hundred keV. The differences come almost entirely from the isovector correlations in the time-odd channel, as shown in the lower panel of Fig. 3 for a representative example of $^{34}$Cl. Let us stress that these poorly-known correlations may significantly impact the ISB corrections, as shown in the upper panel of Fig. 3.

• The type of the isovector time-odd correlations captured by the HF solutions depends on the relative orientation of the nucleonic currents with respect to the nuclear shapes. Solutions oriented perpendicular to the long axis, $\vert\varphi_{\rm X}\rangle$ and $\vert\varphi_{\rm Y}\rangle$, are usually similar to one another (they yield identical correlations for axial systems) and differ from $\vert\varphi_{\rm Z}\rangle$, oriented parallel to the long axis, which captures more correlations due to the current-current time-odd interactions.

• The three $\vert T=1,I=0^+\rangle$ states projected from the $\vert\varphi_{\rm X}\rangle$, $\vert\varphi_{\rm Y}\rangle$, and $\vert\varphi_{\rm Z}\rangle$ Slater determinants differ in energy by only a few tens of keV, see the lower panel of Fig. 3. Hence, energy-wise, they represent the same physical solution, differing only slightly due to the polarization effects originating from different components of the time-odd isovector fields. However, since these correlations are completely absent in the even-even parent nuclei, they strongly impact the calculated $\delta_{\rm C}$. The largest differences in $\delta_{\rm C}$ have been obtained for $A=34$ and $A=74$ systems, see Fig. 3 and Tables 2 and 3.

• Symmetry-unrestricted calculations always converge to the signature-symmetry-conserving solution $\vert\varphi_Z\rangle$ which, rather surprisingly, appears to be energetically unfavored (except for $^{18}$F). In spite of our persistent efforts, no self-consistent tilted-axis solutions have been found.

Figure 3: Upper panel: the ISB corrections $\delta_{\rm C}$ for the $0^+ \rightarrow 0^+$ superallowed $\beta$-decays $^{34}$Ar $\rightarrow ^{34}$Cl (open circles) and $^{34}$Cl $\rightarrow ^{34}$S (full circles) determined for the shape-current orientations $X$, $Y$, and $Z$ depicted schematically in Fig. 2. Lower panel: differences between the energies of the $X$ and $Y$ configurations, and the $Z$ configuration in $^{34}$Cl. Full triangles correspond to the total HF energies and open triangles correspond to contributions from the time-odd isovector channel. Full dots show the total energy differences obtained for the angular-momentum and isospin-projected states.