Decomposition of $d=9$ short-range $0\nu\beta\beta$ decay operators at one-loop level

We perform a systematical study of the dimension-9 short-range $0\nu\beta\beta$ decay operators at one-loop level. There are only six genuine topologies which generate eight diagrams, and the recipe to identify the possible one-loop realizations of the $0\nu\beta\beta$ decay operators is sketched. Certain hypercharge assignments are excluded by the absence of tree-level diagrams in a genuine one-loop model. The mediators of each decomposition can generate Majorana neutrino masses which are discussed up to two-loop level. We present an example of $0\nu\beta\beta$ decay model in which the neutrino masses are generated at two-loop level, and the short-range contribution can be comparable with the mass mechanism in some region of parameter space.