either scalars or vectors, all possible combination in principle can occur (SSS, VVV, VVS and SSV).

One has multiple choices for assigning the fermions to the outer legs. Once a particular assignment is chosen, the electric charge and the colour of the internal particles are fixed (the latter up to a two-fold ambiguity), but not the $U(1)_{Y}$ hypercharge. The assignments of hypercharge are fixed as well, once the chiralities of the six outer fermions are determined. We will come back to this point later in the discussion, and more details are given in the appendix. We will now discuss the general decompositions of T-I and T-II in turn.

Decomposition of topology I
---------------------------

In Tab. \[Tab:TopoI\] we list the general decompositions for the [$0\nu\beta\beta$ ]{}decay operator for T-I. The chiralities of the outer fermions are left unspecified here for a more compact presentation. In total there are 18 (times 2 for the choice of colour) possibilities for realizing T-I. Because of the $SU(3)$ multiplication rules, ${\bf \bar{3}} \otimes {\bf \bar{3}} = {\bf 3}_a \oplus {\bf \bar{6}}_s$ and ${\bf 3} \otimes {\bf \bar{3}} = {\bf 1} \oplus {\bf 8}$, there are always two possible colour assignments for the internal particles. The 