$0\nu \beta \beta$ and $2\nu \beta \beta$ nuclear matrix elements, quasiparticle random-phase approximation, and isospin symmetry restoration

Within the quasiparticle random-phase approximation (QRPA) we achieve partial restoration of the isospin symmetry and hence fulfillment of the requirement that the $2\nu \beta \beta$ Fermi matrix element $M_F^{2\nu }$ vanishes, as it should, unlike in the previous version of the method. This is accomplished by separating the renormalization parameter $g_{pp}$ of the particle-particle proton-neutron interaction into isovector and isoscalar parts. The isovector parameter $g_{pp}^{T=1}$ needs to be chosen to be essentially equal to the pairing constant $g_{\mathrm{pair}}$, so no new parameter is needed. For the $0\nu \beta \beta$ decay the Fermi matrix element $M_F^{0\nu }$ is substantially reduced, while the full matrix element $M^{0\nu }$ is reduced by $\approx$10$\%$. We argue that this more consistent approach should be used from now on in the proton-neutron QRPA and in analogous methods.

Figure 1

Dependence of the $2\nu \beta \beta$ matrix elements $M_F^{2\nu }$ and $M_{\mathrm{GT}}^{2\nu }$ on the isovector coupling constant $g_{pp}^{T=1}$. This example is for ${}^{76}$Ge and ${}^{130}$Te, the Argonne V18 potential, and the isoscalar coupling constants $g_{pp}^{T=0}$ = 0.750 and 0.783.

Figure 2

Functions $C_F^{2\nu }\left(r\right)$ with old and new parametrization and, for comparison, the function $C_{\mathrm{GT}}^{2\nu }\left(r\right)$ (scaled by 1/3 for clarity) is also shown. This is the case of ${}^{76}$Ge.

Figure 3

(a) The function $C_{F\left(cl\right)}^{2\nu }\left(r\right)$ for the pure pairing case, i.e., $g_{pp}^{T=0}=g_{pp}^{T=1}=g_{ph}=0.0$, separated into $S=0$ and $S=1$ components. (b) The function $C_{F\left(cl\right)}^{2\nu }\left(r\right)$ for $g_{pp}^{T=1}$ = 1.038 and $g_{pp}^{T=0}=0.750$ again separated into its $S=0$ and $S=1$ parts. The sum function is also displayed. The dominance of the $S=0$ component is clearly visible in (a). In (b) the two components when integrated over $r$ are, naturally, equal and opposite. The $S=0$ part, however, clearly is considerably larger in absolute value than the $S=1$ part, at all $r$ values. This is the case of ${}^{76}$Ge.

Figure 4

Multipole decomposition of the matrix element $M_F^{0\nu }$. The results with the old and new parametrizations are compared. Note the dominant effect for the $0^{+}$ multipole and the relatively small effects for the other multipoles. This is the case of ${}^{76}$Ge.

Figure 5

Nuclear matrix elements $M^{0\nu }$ evaluated with the new parametrization developed in this work (filled squares) compared with the old method ($g_{pp}^{T=1}=g_{pp}^{T=0}\equiv g_{pp}$) (empty circles). This is a QRPA with $g_A$ = 1.27 and a large-size single-particle level scheme, as in Table , evaluation using the Argonne V18 potential.