(250,10)(290,50) (250,90)(290,50) (290,50)(370,50)[3]{}[8]{} (330,50)(330,90)[3]{}[4]{} (330,90)(290,130) (330,90)(370,130) (370,50)(410,10) (370,50)(410,90) (290,50)[3]{} (330,50)[3]{} (330,90)[3]{} (370,50)[3]{} (295,45)\[t\][$\text v_1$]{} (330,45)\[t\][$\text v_2$]{} (340,85)\[t\][$\text v_3$]{} (370,45)\[t\][$\text v_4$]{} (110,5)\[t\][[Topology I]{}]{} (330,5)\[t\][[Topology II]{}]{} (55,45)\[t\][$\text v_1$]{} (90,45)\[t\][$\text v_2$]{} (130,45)\[t\][$\text v_3$]{} (170,45)\[t\][$\text v_4$]{}

The $d=9$ effective operator generating [$0\nu\beta\beta$ ]{}decay at the quark level can be written as in Eq. . In this section, we decompose this operator, following the techniques developed in [@Gavela:2008ra; @Bonnet:2009ej; @Bonnet:2011yx; @Bonnet:2012kz], in terms of the SM quantum numbers of the mediators. Note that the results obtained in this section are valid for both short- and long-range contributions, while the quantitative impact of the contribution to [$0\nu\beta\beta$ ]{} depends on the Lorentz nature. Details are left for the appendix, where we tabulate all possibilities for the short-range part mediated by fermions and scalars.

At tree-level, there are only two possible topologies for this operator, which are shown in Fig. \[Fig:0nbbTopologies\]. For topology I (T-I) the internal particles between vertices $\text{v}_1-\text{v}_2$ and $\text{v}_3-\text{v}_4$ can be either scalars (S) or vectors (V), while the particle between $\text{v}_2-\text{v}_3$ must be a fermion (F). The topology thus contains three classes of diagrams: VFV, SFS and SFV. The inner particles for topology II (T-II), on the other hand, must all be 