Seesaw mechanisms for Dirac and Majorana neutrino masses

We investigate the seesaw mechanism for generally non-fine-tuned $n\times n$ mass matrices involving both Dirac and Majorana neutrinos. We specifically show that the number of naturally light neutrinos cannot exceed half of the dimension of the considered mass matrix. Furthermore, we determine a criterion for mass matrix textures leading to light Dirac neutrinos with the seesaw mechanism. In particular, we study $4\times 4$ and $6\times 6$ mass matrix textures and give some examples in order to highlight these types of texture. Next, we present a model scheme based on non-Abelian and discrete symmetries satisfying the above mentioned criterion for light Dirac neutrinos. Finally, we investigate the connection between symmetries and the invariants of a mass matrix on a formal level.