We use diagrammatic many-body perturbation theory in combination with low-momentum interactions derived from chiral effective field theory to construct effective shell-model transition operators for the neutrinoless double-$\beta$ decay of ${}^{76}$Ge and ${}^{82}$Se. We include all unfolded diagrams that are first and second order in the interaction and all singly folded diagrams that can be constructed from them. The resulting effective operator, which accounts for physics outside the shell-model space, increases the nuclear matrix element by about 20$\%$ in ${}^{76}$Ge and 30$\%$ in ${}^{82}$Se.

• Received 16 April 2013

DOI:https://doi.org/10.1103/PhysRevC.87.064315