What if a specific neutrinoless double beta decay is absent

We consider the seesaw model with two right-handed neutrinos $N_1$ and $N_2$ which masses are hierarchical, and investigate their contribution to the neutrinoless double beta ($0 \nu \beta \beta$) decay. Although the lepton number is broken by the Majorana masses of right-handed neutrinos, such decay processes can be absent in some cases. We present a possibility where the lighter $N_1$ gives a destructive contribution to that of active neutrinos by choosing the specific mixing elements of $N_1$, while $N_2$ is sufficiently heavy not to contribute to the $0 \nu \beta \beta$ decay. In this case the mixing elements of $N_1$ in the charged current interaction are determined by its mass and the Majorana phase of active neutrinos. We then study the impacts of such a possibility on the direct search for $N_1$. In addition, we discuss the consequence of the case when the $0 \nu \beta \beta$ decay in one specific nucleus is absent.