Nonperturbative renormalization of the neutrinoless double-$\beta$ operator in $p$-shell nuclei

We use Lee-Suzuki mappings and related techniques to construct effective two-body $p$-shell interactions and neutrinoless double-$\beta$ operators that exactly reproduce the results of large no-core-shell-model calculations of (fictitious) double-$\beta$ decay in nuclei with mass number $A=6$. We then apply the effective operators to the (also fictitious) decay of nuclei with $A=7$, 8, and 10, again comparing with no-core calculations in much larger spaces. The results with the effective two-body operators are generally good. In some cases, however, they differ non-negligibly from the full no-core results, suggesting that three-body corrections to the decay operator in heavier nuclei may be important. An application of our procedure and related ideas to $fp$-shell nuclei such as ${}^{76}$Ge should be feasible within coupled-cluster theory.

Figure 1

The curves $C\left(r\right)$, the integrals of which give the matrix elements for neutrinoless double-$\beta$ decay. The solid (black) lines are the results of the full (${\mathrm{N}}_{\max }=6$) calculations with the SRG-evolved N${}^3$LO potential, the dashed (red) lines are the results of the $p$-shell calculation with the effective two-body Hamiltonian and the bare decay operator, and the dot-dashed (blue) lines are the results with the effective Hamiltonian and the effective decay operator. The top panel is for the decay ${}^7$He $\to {}^7$Be, the middle panel for ${}^8$He $\to {}^8$Be, and the bottom panel for ${}^{10}$He $\to {}^{10}$Be.

Figure 2

Same as Fig. 1 but for for ${}^8$He $\to {}^8$Be and ${}^{10}$He $\to {}^{10}$Be only and with the CD-Bonn interaction conditioned for ${\mathrm{N}}_{\max }=8$ in $A=8$ and ${\mathrm{N}}_{\max }=6$ in $A=10$.

Figure 3

Same as Fig. 1 but carried out in a small model space with $N_{\max }=2$.

Figure 4

$C\left(r\right)$ for ${}^8$He ${\to }^8$Be: the solid (black), dashed (red), and dot-dashed (blue) lines are as in Fig. 1. The dotted (maroon) and the dot-dot-dashed (green) curves result from using the normalized $p$-shell projections of the full ground states with the effective and the bare decay operator respectively. The dot-dashed-dashed (orange) curve is the result with bare decay operator and the Cohen-Kurath (CK) interaction [25].