Large-Scale Shell-Model Analysis of the Neutrinoless $\beta \beta$ Decay of ${}^{48}\mathrm{Ca}$

We present the nuclear matrix element for the neutrinoless double-beta decay of ${}^{48}\mathrm{Ca}$ based on large-scale shell-model calculations including two harmonic oscillator shells ($sd$ and $pf$ shells). The excitation spectra of ${}^{48}\mathrm{Ca}$ and ${}^{48}\mathrm{Ti}$, and the two-neutrino double-beta decay of ${}^{48}\mathrm{Ca}$ are reproduced in good agreement to the experimental data. We find that the neutrinoless double-beta decay nuclear matrix element is enhanced by about 30% compared to $pf$-shell calculations. This reduces the decay lifetime by almost a factor of 2. The matrix-element increase is mostly due to pairing correlations associated with cross-shell $sd\text{−}pf$ excitations. We also investigate possible implications for heavier neutrinoless double-beta decay candidates.

Figure 1

Excitation spectra of ${}^{48}\mathrm{Ca}$ and ${}^{48}\mathrm{Ti}$. The lowest five positive-parity states [41] are compared to $sdpf$ calculations with the SDPFMU-DB interaction.

Figure 2

Comparison of NME values for the ${}^{48}\mathrm{Ca}$ $0\nu \beta \beta$ decay. The present shell-model results in the $sdpf$ space (SM $sdpf$: left SDPFMU-DB, right SDPFMU) are compared to $pf$-shell results (SM $pf$: left [17], right [15]), $pf$-shell result plus a perturbative calculation of the effect of orbitals outside the $pf$ shell (SM MBPT) [50], QRPA [22], IBM [25], and EDF (left: nonrelativistic [26], right: relativistic [27]) calculations. The range between double horizontal bars covers results including a different type of short-range correlations (Argonne, CD-Bonn, UCOM [51]) and without them.

Figure 3

NME decomposition in terms of the angular momentum and parity $J^{\pi }$ of the pair of decaying neutrons, Eq. (3). $0\hslash \omega$ (GXPF1B) and $2\hslash \omega$ (SDPFMU-DB) results are compared, without short-range correlations.

Figure 4

Diagrams associated with the NME decomposition in Eq. (4), classified in terms of the $sd$- or $pf$-shell orbitals occupied by the decaying neutrons (open circles) and created protons (filled circles). Initial (final) stands for ${}^{48}\mathrm{Ca}$ (${}^{48}\mathrm{Ti}$). Diagrams (i)–(v) correspond to ${\mathcal{M}}_1^{0\nu }-{\mathcal{M}}_5^{0\nu }$, respectively.