Correlations and neutrinoless $\beta \beta$ decay nuclear matrix elements of $pf$-shell nuclei

We calculate the nuclear matrix elements (NMEs) for neutrinoless double-beta decays of $pf$-shell nuclei using the shell model (SM) and energy density functional (EDF) methods. The systematic study of the decays of $\mathrm{Ca}\to \mathrm{Ti}$, $\mathrm{Ti}\to \mathrm{Cr}$ and $\mathrm{Cr}\to \mathrm{Fe}$ isotopes allows for a detailed comparison between the two nuclear structure many-body approaches. We observe that while the dominant Gamow-Teller part of the NME differs roughly by a factor of 2 between SM and EDF, when we restrict the calculations to spherical EDF states and seniority-zero SM configurations, the NMEs obtained by both methods are strikingly similar. This points out the important role of nuclear structure correlations for neutrinoless double-beta decay NMEs. We identify correlations associated with high-seniority components in the initial and final states of the decay as one of the reasons for the discrepancies between SM and EDF results. We also explore exact projection to good isospin, and conclude that it strongly reduces the Fermi contribution of the NMEs, but it has only a moderate effect in the Gamow-Teller part. This work opens up the door for NME benchmarks between different theoretical approaches, and constitutes a step forward towards more reliable estimations of the NMEs.

Figure 1

Gamow-Teller part of the nuclear matrix element, $M_{\mathrm{GT}}^{0\nu }$, for $\mathrm{Ca}\to \mathrm{Ti}$ (a), $\mathrm{Ti}\to \mathrm{Cr}$ (b) and $\mathrm{Cr}\to \mathrm{Fe}$ (c) $0\nu \beta \beta$ decays, calculated with shell model (SM) and energy density functional (EDF) methods. The D1S EDF interaction is used (circles). In the SM case, the KB3G (squares) and GXPF1A (lozenges) effective interactions are employed.

Figure 2

Gamow-Teller part of the nuclear matrix element, $M_{\mathrm{GT}}^{0\nu }$, for $\mathrm{Ca}\to \mathrm{Ti}$ (a), $\mathrm{Ti}\to \mathrm{Cr}$ (b), and $\mathrm{Cr}\to \mathrm{Fe}$ (c) $0\nu \beta \beta$ decays, with seniority-zero shell model (SM) and spherical energy density functional (EDF) states. Interactions are as in Fig. 1.

Figure 3

(a) Particle-number and angular-momentum projected $\left(J=0\right)$ potential energy surfaces (thin lines) and ground-state collective wave functions (thick lines) for ${}^{58}\mathrm{Ti}$ (dashed) and ${}^{58}\mathrm{Cr}$ (solid) nuclei as a function of the quadrupole deformation ${\beta }_2$. Triangles (squares) correspond to the spherical points (minima) of the corresponding surfaces. (b)–(d) Gamow-Teller part of the nuclear matrix element, $M_{\mathrm{GT}}^{0\nu }$, for (b) $\mathrm{Ca}\to \mathrm{Ti}$, (c) $\mathrm{Ti}\to \mathrm{Cr}$, and (d) $\mathrm{Cr}\to \mathrm{Fe}$ (d) $0\nu \beta \beta$ decays. Calculations use the Gogny D1S functional with initial and final states obtained at the level of spherical calculation (red triangles), taking the minimum of the potential-energy surface in a deformed calculation (blue squares), and in the full calculation with configuration mixing (black circles); see panel (a) for the different approaches.

Figure 4

Shell model results for the Fermi (left panels) and Gamow-Teller (right panels) parts of the nuclear matrix element, $M_F^{0\nu }$ and $M_{\mathrm{GT}}^{0\nu }$, for $\mathrm{Ca}\to \mathrm{Ti}$ (a)-(b), $\mathrm{Ti}\to \mathrm{Cr}$ (c)-(d) and $\mathrm{Cr}\to \mathrm{Fe}$ (e)-(f) $0\nu \beta \beta$ decays. Calculations are performed—using KB3G interaction—with initial and final states obtained at zero seniority (red diamonds), zero-seniority with exact isospin projection (blue inverted triangles), and in the full calculation (black left triangles).

Figure 5

Gamow-Teller $[M_{\mathrm{GT}}^{0\nu }$, panels (a),(b)] and Fermi $[M_{\mathrm{F}}^{0\nu }$, panels (c),(d)] parts of the nuclear matrix element of the $0\nu \beta \beta$ decays of ${}^{50}\mathrm{Ca}{\to }^{50}\mathrm{Ti}$ [panels (a),(c)] and ${}^{48}\mathrm{Ti}{\to }^{48}\mathrm{Cr}$ [panels (b),(d)]. Shell model (SM) results are shown as a function of the maximum seniority permitted in the initial and final states (squares), and also after isospin projection (circles). Energy density functional (EDF) results using spherical initial and final states (dashed lines) and the full EDF calculation (dashed-dotted lines) are also shown. The EDF Gogny D1S and SM KB3G interactions are used.