Partial restoration of spin-isospin SU(4) symmetry and the one-quasiparticle random-phase approximation method in double-$\beta$ decay

The one-quasiparticle random-phase approximation (one-QRPA) method is used to describe simultaneously both double-$\beta$-decay modes, giving special attention to the partial restoration of spin-isospin $\text{SU}\left(4\right)$ symmetry. To implement this restoration and to fix the model parameters, we resort to the energetics of Gamow-Teller resonances and to the minima of the single-${\beta }^{+}$-decay strengths. This makes the theory predictive regarding the $\beta {\beta }_{2\nu }$ decay, producing the $2\nu$ moments in ${}^{48}\mathrm{Ca}$, ${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, ${}^{96}\mathrm{Zr}$, ${}^{100}\mathrm{Mo}$, ${}^{128,130}\mathrm{Te}$, and ${}^{150}\mathrm{Nd}$, that are of the same order of magnitude as the experimental ones; however, the agreement with $\beta {\beta }_{2\nu }$ data is only modest. To include contributions coming from induced nuclear weak currents, we extend the $\beta {\beta }_{0\nu }$-decay formalism employed previously in C. Barbero et al., Nucl. Phys. A 628, 170 (1998), which is based on the Fourier-Bessel expansion. The numerical results for the $\beta {\beta }_{0\nu }$ moments in the above mentioned nuclei are similar to those obtained in other theoretical studies although smaller on average by $\sim 40\%$. We attribute this difference basically to the one-QRPA method, employed here for the first time, instead of the currently used two-QRPA method. The difference is partially due also to the way of carrying out the restoration of the spin-isospin symmetry. It is hard to say which is the best way to make this restoration, since the $\beta {\beta }_{0\nu }$ moments are not experimentally measurable. The recipe proposed here is based on physically robust arguments. The numerical uncertainties in the $\beta \beta$ moments, related to (i) their strong dependence on the residual interaction in the particle-particle channel when evaluated within the QRPA, and (ii) lack of proper knowledge of single-particle energies, have been quantified. It is concluded that the partial restoration of the $\text{SU}\left(4\right)$ symmetry, generated by the residual interaction, is crucial in the description of the $\beta \beta$ decays, regardless of the nuclear model used.

Figure 1

(a) and ${(\mathrm{a}}^{\prime }$): $0\nu$ NM normalized to $g_A^2$; (b) and ${(\mathrm{b}}^{\prime }$): $2\nu$ NM given in natural units; and (c) and ${(\mathrm{c}}^{\prime }$): ${\beta }^{+}$-decay transition strengths. We show the vector observables, as a function of the ratio $s={\mathit{v}}_{\text{pp}}^s/{\overline{v}}_{\mathrm{pair}}^s$, on the left side, and axial-vector ones, as a function of the ratio $t={\mathit{v}}_{\text{pp}}^t/{\overline{v}}_{\mathrm{pair}}^s$, on the right side. The values of $t_{\mathrm{sym}}$ on the axis $t$ are indicated by points.

Figure 2

Isoscalar parameters $t$ in ${}^{76}\mathrm{Ge}$ within method II for $|M_{exp}^{2\nu }|=0.113$. In (a) the NM $M^{2\nu }$ is given in natural units, while in (b) the $M^{0\nu }$ is dimensionless. It should be noted that $M^{2\nu }$ is negative at $t=0$.

Figure 3

$\beta {\beta }_{0\nu }$ nuclear moments evaluated with several nuclear structure model calculations: (i) QRPA by Tübingen (QRPA Tü) [9] ($g_A=1.27$), Jyväskylä (QRPA Jy) [11] ($g_A=1.26$) groups, and our results from Table 3 (QRPA Ferr) ($g_A=1.27$); (ii) interacting shell model (ISM) [54] ($g_A=1.25$) and large-scale shell model [SM (SDPFMU)] [62]; (iii) interacting boson model (IBM-2) [63] ($g_A=1.269$); and (vi) energy density functional method (EDF) [64] ($g_A=1.25$) and covariant density functional theory (CDFT) [35, 65] ($g_A=1.254$). All results are normalized to $g_A^2$. The theoretical “errors” presented in our calculations are evaluated as described in the text. The present figure is similar to [35, Fig. 7], [66, Fig. 5], [67, Fig. 4], and [68, Fig. 1].