Shell-model calculations of two-neutrino double-β decay rates of ${}^{48}\mathrm{Ca}$ with the GXPF1A interaction

The two-neutrino double-β decay matrix elements and half-lives of ${}^{48}\mathrm{Ca}$ were calculated within a shell-model approach for transitions to the ground state and to the $2^{+}$ first excited state of ${}^{48}\mathrm{Ti}$. We use the full $\mathit{pf}$ model space and the GXPF1A interaction, which was recently proposed to describe the spectroscopic properties of the nuclei in the nuclear mass region $A=47$–66. Our results are $T_{1/2}(0^{+}\to 0^{+})=3.3\times {10}^{19}$ yr and $T_{1/2}(0^{+}\to 2^{+})=8.5\times {10}^{23}$ yr. The result for the decay to the ${}^{48}\mathrm{Ti}$ $0^{+}$ ground state is in good agreement with experiment. The half-life for the decay to the $2^{+}$ state is two orders of magnitude larger than that obtained previously.

Figure 1

The $2\nu \beta \beta$ decay running matrix element of Eq. (1) as a function of the excitation energy of the intermediate $1^{+}$ states in ${}^{48}\mathrm{Sc}$ for the GXPF1 and GXPF1A interactions.

Figure 2

The $2\nu \beta \beta$ decay running matrix element of Eq. (2) (up to a factor of $\sqrt{5}$) as a function of the excitation energy of the intermediate $1^{+}$ states in ${}^{48}\mathrm{Sc}$ for the GXPF1 and GXPF1A interactions.

Figure 3

The $2\nu \beta \beta$ decay running matrix element of Eq. (1) as a function of the excitation energy of the intermediate $1^{+}$ states in ${}^{48}\mathrm{Sc}$ compared with the similar sum where the absolute values of the GT matrix elements are used.

Figure 4

The relevant $B$(GT) probabilities for transitions to the $1^{+}$ states in ${}^{48}\mathrm{Sc}$.