Systematic study of one-loop realizations of $d=7$ long-range $0\nu \beta \beta$ decay operators

We study the systematical one-loop decomposition of the dimension-7 long-range $0\nu \beta \beta$ decay operators. We find that there are 3 genuine one-loop topologies and 8 diagrams. The procedure to determine the SM quantum number assignments for both internal and external fields is presented. The Majorana neutrino mass in long-range $0\nu \beta \beta$ models is discussed. We also present a one-loop $0\nu \beta \beta$ decay model which produces Majorana neutrino mass at three-loop level. The phenomenological predictions for light neutrino mass and $0\nu \beta \beta$ decay half-life time including both mass mechanism and long-range contribution are studied.