ISB corrections to the Fermi matrix elements in mirror-symmetric $\bm {T=1/2}$ nuclei

Transitions between the isobaric analogue states in mirror nuclei $\vert T=1/2, I, T_z= -1/2\rangle$ $\rightarrow$ $\vert T=1/2, I, T_z= + 1/2\rangle$ offer an alternative way to extract the ${\cal F}t$-values [49] and $V_{ud}$ [40,50]. Those transitions are mixed Fermi and Gamow-Teller, meaning that they are mediated by both the vector and axial-vector currents. Hence, the extraction of $V_{ud}$ requires - in addition to lifetimes and $Q$-values - measuring another observable, such as the beta-neutrino correlation coefficient, beta-asymmetry, or neutrino-asymmetry parameter [51,52]. Moreover, the method depends on the radiative and ISB corrections to both the Fermi and Gamow-Teller matrix elements. In spite of these difficulties, current precision of determination of $V_{ud}$ using the mirror-decay approach is similar to that offered by neutron-decay experiments [10,40,50], see also Figs. 8 and 9.

Within our projected-DFT model, we performed systematic calculations of ISB corrections to the Fermi matrix elements, $\delta_{\text{C}}^{\text{V}}$, covering the mirror transitions in all $11\leq A \leq 49$ nuclei. Calculations were based on the Slater determinants corresponding to the lowest-energy, unrestricted-symmetry HF solutions. If the unrestricted-symmetry calculations did not converge, the projection was applied to the constrained HF solutions with imposed signature symmetry. These two types of solutions differ, in particular, in relative shape-current orientation, which also varies with $A$ depending on the s.p. orbit occupied by an unpaired nucleon. It should be underlined, however, that the HF solutions corresponding to the $\beta$-decay partners were always characterized by the same orientation of the odd-particle alignment with respect to the body-fixed reference frame. All calculations discussed in this section were performed by using the full basis of $N=12$ HO shells and the SV force.

Table 5: Results of calculations for $\vert T=1/2, I, T_z= -1/2\rangle$ $\rightarrow$ $\vert T=1/2, I, T_z= + 1/2\rangle$ $\beta$-decays between mirror nuclei: theoretical spin and parity assignments; isospin-mixing coefficients in the parent and daughter nuclei; ISB corrections calculated in this work (asterisks denote results obtained within unrestricted-symmetry calculations); ISB corrections of Ref. [49]; quadrupole equilibrium deformation parameters in the parent nuclei; and theoretical and experimental $Q_\beta$ values.

			$\quad$	$I^\pi$	$\quad$	$\alpha_{\rm C}^{{\rm (P)}}$	$\alpha_{\rm C}^{{\rm (D)}}$	$\quad$	$\delta^{{\rm (SV)}}_{\rm C}$	$\delta_{\rm C}^{{\rm (S)}}$	$\quad$	$\beta_2^{{\rm (SV)}}$	$\gamma^{{\rm (SV)}}$	$\quad$	$Q_\beta^{{\rm (th)}}$	$Q_\beta^{{\rm (exp)}}$
			$\quad$		$\quad$	(%)	(%)	$\quad$	(%)	(%)	$\quad$		(deg)	$\quad$	(MeV)	(MeV)
$^{11}$C	$\rightarrow$	$^{11}$B		$\frac{3}{2}^-$		0.001	0.003		0.077	0.928		0.320	43.8		1.656	1.983
$^{13}$N	$\rightarrow$	$^{13}$C		$\frac{1}{2}^-$		0.008	0.001		0.139	0.271		0.210	59.1		1.888	2.221
$^{15}$O	$\rightarrow$	$^{15}$N		$\frac{1}{2}^-$		0.012	0.002		0.127	0.181		0.003	0.0		2.446	2.754
$^{17}$F	$\rightarrow$	$^{17}$O		$\frac{5}{2}^+$		0.020	0.031		0.167	0.585		0.014	0.0		2.496	2.761
						0.019	0.029		$^*$0.178	0.585		0.064	60.0		2.499
$^{19}$Ne	$\rightarrow$	$^{19}$F		$\frac{1}{2}^+$		0.036	0.034		0.365	0.415		0.321	0.0		2.928	3.239
$^{21}$Na	$\rightarrow$	$^{21}$Ne		$\frac{3}{2}^+$		0.047	0.052		0.307	0.348		0.434	0.0		3.229	3.548
$^{23}$Mg	$\rightarrow$	$^{23}$Na		$\frac{3}{2}^+$		0.064	0.070		0.340	0.293		0.434	0.0		3.587	4.057
$^{25}$Al	$\rightarrow$	$^{25}$Mg		$\frac{5}{2}^+$		0.073	0.058		0.503	0.461		0.444	1.6		3.683	4.277
$^{27}$Si	$\rightarrow$	$^{27}$Al		$\frac{5}{2}^+$		0.074	0.073		0.472	0.312		0.343	47.7		4.250	4.813
$^{29}$P	$\rightarrow$	$^{29}$Si		$\frac{1}{2}^+$		0.123	0.113		0.694	0.976		0.332	54.4		4.399	4.943
$^{31}$S	$\rightarrow$	$^{31}$P		$\frac{5}{2}^+$		0.163	0.164		0.504	0.715		0.315	0.0		4.855	5.396
$^{33}$Cl	$\rightarrow$	$^{33}$S		$\frac{3}{2}^+$		0.177	0.160		0.644	0.865		0.258	33.5		5.002	5.583
$^{35}$Ar	$\rightarrow$	$^{35}$Cl		$\frac{3}{2}^+$		0.186	0.182		0.576	0.493		0.209	50.4		5.482	5.966
$^{37}$K	$\rightarrow$	$^{37}$Ar		$\frac{3}{2}^+$		0.291	0.267		1.425	0.734		0.143	60.0		5.589	6.149
$^{39}$Ca	$\rightarrow$	$^{39}$K		$\frac{3}{2}^+$		0.318	0.289		$^*$0.392	0.855		0.034	60.0		6.084	6.531
$^{41}$Sc	$\rightarrow$	$^{41}$Ca		$\frac{7}{2}^-$		0.341	0.345		$^*$0.426	0.821		0.032	60.0		5.968	6.496
$^{43}$Ti	$\rightarrow$	$^{43}$Sc		$\frac{7}{2}^-$		0.376	0.380		$^*$0.463	0.500		0.090	60.0		6.225	6.868
$^{45}$V	$\rightarrow$	$^{45}$Ti		$\frac{7}{2}^-$		0.437	0.424		0.534	0.865		0.233	0.0		6.563	7.134
						0.438	0.427		$^*$0.661	0.865		0.233	0.0		6.559
$^{47}$Cr	$\rightarrow$	$^{47}$V		$\frac{3}{2}^-$		0.480	0.457		0.518	--		0.276	0.0		6.827	7.452
						0.483	0.463		$^*$0.710	--		0.275	0.0		6.826
$^{49}$Mn	$\rightarrow$	$^{49}$Cr		$\frac{5}{2}^-$		0.515	0.497		0.522	--		0.284	0.9		7.054	7.715
						0.518	0.499		$^*$0.681	--		0.284	0.0		7.053

The obtained values of the ISB corrections to the Fermi transitions,

$$\delta_{\text{C}}^{\text{V}} \equiv 1- \vert \langle T=\frac{1}{2... ...2} \vert \hat T_\mp \vert T= \frac{1}{2},I,T_z=\pm \frac{1}{2} \rangle \vert^2,$$ (25)

are collected in Table 5 and illustrated in Fig. 13.

Figure 13: Full circles: calculated values of the ISB corrections to the Fermi transitions in $T=1/2$ mirror nuclei. Open circles with errors: results calculated by Severijns et al. [49].

Since the calculations were performed in a relatively large basis, the basis-cut-off-related uncertainty in $\delta_{\text{C}}^{\text{V}}$ could be reduced to approximately $5\%$, cf. Sec. 3.3. Except for one case, theoretical spins and parities of decaying states were taken equal to those found in experiment: $I^\pi_{\text{(th)}}=I^\pi_{{\text{(exp)}}}$(g.s.). Only for $A=31$, no $I=1/2$ component was found in the HF wave function, and thus the lowest solution corresponding to $I^\pi_{\text{(th)}}=5/2^+$ was taken instead. It should be mentioned that, owing to the poor spectroscopic quality of SV, the projected states corresponding to $I^\pi_{{\text{(exp)}}}$(g.s.) are not always the lowest ones. This situation occurs for $A = 19$, 25, and 45, where the lowest states have $I^\pi_{\text{(th)}}=5/2^+$, $1/2^+$, and $3/2^-$, and the corresponding $\delta_{\text{C}}^{\text{V}}$ values are 0.308 %, 0.419%, and 0.636%, respectively. A relatively strong dependence of the calculated ISB corrections on spin is worth noting. The calculations also indicate an appreciable impact of the signature-symmetry constraint on $\delta_{\text{C}}^{\text{V}}$, in particular, in the $pf$-shell nuclei with $A=45$, 47, and 49. A similar effect was calculated for the $0^+ \rightarrow 0^+$ transitions, see $\delta_{\rm C}$-values at fixed shape-current orientations in Tables 2 and 3.