ISB corrections to the superallowed $\bm {0^+ \rightarrow 0^+}$ Fermi transitions

The $0^+ \rightarrow 0^+$ Fermi $\beta$-decay proceeds 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, $\vert I=0, T\approx 1, T_z = 0 \rangle$. The corresponding transition matrix element is:

$$M_{\rm F}^{(\pm )} = \langle I=0, T\approx 1, T_z = \pm 1 \vert \hat T_{\pm} \vert I=0, T\approx 1, T_z = 0 \rangle.$$ (17)

The g.s. state $\vert I=0, T\approx 1, T_z = \pm 1 \rangle$ in Eq. (17) is approximated by a projected state

$$\vert I=0, T\approx 1, T_z = \pm 1 \rangle = \sum_{T\geq 1} c^{( \psi )}_{T} \hat P^T_{\pm 1, \pm 1} \hat P^{I=0}_{0,0} \vert\psi \rangle,$$ (18)

where $\vert\psi \rangle$ is the g.s. of the even-even nucleus obtained in self-consistent MF calculations. The state $\vert\psi \rangle$ is unambiguously defined by filling in the pairwise doubly degenerate levels of protons and neutrons up to the Fermi level. The daughter state $\vert I=0, T\approx 1, T_z = 0 \rangle$ is approximated by

$$\vert I=0, T\approx 1, T_z = 0 \rangle = \sum_{T\geq 0} c^{( \varphi )}_{T} \hat P^T_{0, 0} \hat P^{I=0}_{0,0} \vert\varphi \rangle,$$ (19)

where the self-consistent Slater determinant $\vert\varphi \rangle \equiv \vert\bar \nu \otimes \pi \rangle$ (or $\vert \nu \otimes \bar \pi \rangle$) represents the so-called anti-aligned configuration, selected by placing the odd neutron and the odd proton in the lowest available time-reversed (or signature-reversed) s.p. orbits. The s.p. configuration $\vert\bar \nu \otimes \pi \rangle$ manifestly breaks the isospin symmetry as schematically depicted in Fig. 1. The isospin projection from $\vert\varphi \rangle$ as expressed by Eq. (19) is essentially the only way to reach the $\vert T\approx 1, I=0\rangle$ states in odd-odd $N=Z$ nuclei.

Figure 1: Left: two possible g.s. configurations of an odd-odd $N$=$Z$ nucleus, as described by the conventional deformed MF theory. These degenerate configurations are called aligned (upper) and anti-aligned (lower), depending on what levels are occupied by the valence particles. The right panel shows what happens when the isospin-symmetry is restored. The aligned configuration is isoscalar; hence, it is insensitive to the isospin projection. The anti-aligned configuration represents a mixture of $T$=0 and $T$=1 states. The isospin projection removes the degeneracy by lowering the $T$=0 level.