Neutrinoless double-$\beta$ decay matrix elements in light nuclei

We present the first ab initio calculations of neutrinoless double-$\beta$ decay matrix elements in $A=6-12$ nuclei using variational Monte Carlo wave functions obtained from the Argonne $v_{18}$ two-nucleon potential and Illinois-7 three-nucleon interaction. We study both light Majorana neutrino exchange and potentials arising from a large class of multi-TeV mechanisms of lepton-number violation. Our results provide benchmarks to be used in testing many-body methods that can be extended to the heavy nuclei of experimental interest. In light nuclei we also study the impact of two-body short-range correlations and the use of different forms for the transition operators, such as those corresponding to different orders in chiral effective theory.

Figure 1

Diagrams illustrating the $0\nu \beta \beta$ potentials mediated by neutrinos—$V_{\nu }$ defined in Eq. (6)—and two-pion-exchange, one-pion-exchange, and short-distance interactions—$V_{\pi \pi },V_{\pi N}$, and $V_{NN}$ defined in Eqs. (12).

Figure 2

VMC calculations of the transition densities associated with the F, GT, and T operators—${\sum }_{a<b}\left({\tau }_a^{+}{\tau }_b^{+}\right)$, ${\sum }_{a<b}({\mathbf{\sigma }}_a·{\mathbf{\sigma }}_b{\tau }_a^{+}{\tau }_b^{+})$, and ${\sum }_{a<b}\left(S_{ab}{\tau }_a^{+}{\tau }_b^{+}\right)$, respectively—for the ${}^6\mathrm{He}\to {}^6\mathrm{Be}$ (left panel) and ${}^{10}\mathrm{He}\to {}^{10}\mathrm{Be}$ decays (right panel).

Figure 3

VMC calculations of the transition distributions $C^{\alpha ,\beta }\left(r\right)$ defined in Eq. (26) for the ${}^6\mathrm{He}\to {}^6\mathrm{Be}$ decay.

Figure 4

VMC calculations of the transition distributions $C^{\alpha ,\beta }\left(r\right)$ defined in Eq. (26) for the ${}^{12}\mathrm{Be}\to {}^{12}\mathrm{C}$ decay.

Figure 5

The GT-$\nu$, F-$\nu$, GT-$\pi \pi$, and GT-$\pi N$ distributions in momentum space for the ${}^{10}\mathrm{He}\to {}^{10}\mathrm{Be}$ and ${}^{12}\mathrm{Be}\to {}^{12}\mathrm{C}$ decays. Solid and dashed lines are obtained, respectively, with and without the inclusion of the momentum dependence in nucleonic form factors. See text for explanation.

Figure 6

The left (right) panel shows the GT-AA distribution in $r$-space ($q$-space) for the ${}^{10}\mathrm{He}\to {}^{10}\mathrm{Be}$ transition, with and without “one-pion-exchange-like” correlations in the nuclear wave functions. See text for explanation.