Coupled-Cluster Calculations of Neutrinoless Double-$\beta$ Decay in ${}^{48}\mathrm{Ca}$

We use coupled-cluster theory and nuclear interactions from chiral effective field theory to compute the nuclear matrix element for the neutrinoless double-$\beta$ decay of ${}^{48}\mathrm{Ca}$. Benchmarks with the no-core shell model in several light nuclei inform us about the accuracy of our approach. For ${}^{48}\mathrm{Ca}$ we find a relatively small matrix element. We also compute the nuclear matrix element for the two-neutrino double-$\beta$ decay of ${}^{48}\mathrm{Ca}$ with a quenching factor deduced from two-body currents in recent ab initio calculation of the Ikeda sum rule in ${}^{48}\mathrm{Ca}$ [Gysbers et al., Nat. Phys. 15, 428 (2019)].

Figure 1

Comparison of the NME for the $0\nu \beta \beta$ decay of ${}^{48}\mathrm{Ca}$, calculated within various approaches (see text for details). The coupled-cluster results use both the CCSD and CCSDT-1 approximations with both the spherical and deformed reference states. For $\mathrm{IMSRG}+\mathrm{GCM}$, the double bars show the effects of uncertainty in model-space size; otherwise they show those of uncertainty in short-range correlation functions.

Figure 2

Comparison of the $0\nu \beta \beta$ NME in several light nuclei computed with the coupled cluster method and the no-core shell model. The first two columns correspond to different choices for the coupled-cluster reference state, and results from the CCSD and CCSDT-1 approximations are shown in each. The error bars indicate the uncertainties coming from variations with model-space size. Each case utilizes the $1.8/2.0$ (EM) interaction except for ${}^{22}\mathrm{O}\to {}^{22}\mathrm{Ne}$ which disregards the three-nucleon forces to more rapidly converge the NCSM results.

Figure 3

The NME for the $2\nu \beta \beta$ decay ${}^{48}\mathrm{Ca}\to {}^{48}\mathrm{Ti}$ computed with the $1.8/2.0$ (EM) interaction as a function of the energy difference, $E_I-E_F$, and the $3p-3h$ truncation used to calculate ${}^{48}\mathrm{Sc}$, ${\mathcal{E}}_3$, at $N_{\max }=10$. The results for $N_{\max }=6$, 8 are also shown. The experimental NME and energy difference are shown along with the computed energy difference and NME, with and without a quenching factor of $0.81^2$ deduced from two-body currents [95].