with fixed chiralities.

Much fewer models with T-II have been studied in the literature than for T-I. In fact, we have found only the cases T-II-1 and T-II-5 (with a scalar ${\bf\bar 6}$) have been considered previously. The best known is the case T-II-1 with bosons transforming as $V(+1,{\bf 1})$. This diagram can be produced by adding an $SU(2)_L$ triplet scalar to the SM particle content [@Mohapatra:1981pm] (case SSS).[^12] The same diagram can occur in LR-symmetric models [@Rizzo:1982kn] (as VVS). In this case, the outer fermions are again all right-handed, and this possibility has been considered in [@Hirsch:1996qw]. Note that in all these models, there is always an additional contribution from T-I, which is not necessarily present for all T-II models. In the case of the LR-symmetric model, T-II can be comparable with T-1, but can never completely dominate the contributions to [$0\nu\beta\beta$ ]{}decay.

Finally there is the recent paper [@Gu:2011ak], in which the author constructs a model with the particle content corresponding to the scalars in T-II-5 (with ${\bf\bar 6}$ representation under $SU(3)_{c}$). In this model neutrinos are pseudo-Dirac, such that T-II-5 gives the dominant contribution to [$0\nu\beta\beta$ ]{}decay. This is the only example in the literature, which we 