Neutrinoless double $\beta$ decay and leptogenesis in seesaw model

The fermion ${\nu }_R+C{\overline{{\nu }_R}}^T$ in the Type I seesaw model, which is commonly identified with a Majorana fermion, is not a Majorana fermion and thus no neutrinoless double $\beta$ decay (in the case of ${\nu }_L-C{\overline{{\nu }_L}}^T$) due to the theorem on the absence of a Majorana-Weyl fermion in the representation of the d=4 Lorentz group. The leptogenesis is realized naturally together with the neutrinoless double $\beta$ decay by first defining C symmetric Majorana fermions in the seesaw model using an analogue of the Bogoliubov transformation.